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Federal Information Processing Standards (FIPS) Publication 186-4
Digital Signature Standard (DSS)
July 19, 2013
February 3, 2024
FIPS 186-4 is withdrawn and superseded in its entirety by FIPS 186-5, one year
after the publication of FIPS 186-5 (per its Implementation Schedule clause[12])

Superseding Publication(s) (if applicable)
The attached publication has been superseded by the following publication(s):Series/Number
Title
Author(s)Publication Date(s)URL/DOIFIPS 186-5
Digital Signature Standard (DSS)
National Institute of Standards and Technology
February 3, 2023
https://doi.org/10.6028/NIST.FIPS.186-5

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Date updated: February 5, 2024

FIPS PUB 186-4
FEDERAL INFORMATION PROCESSING STANDARDSPUBLICATION

Digital Signature Standard (DSS)CATEGORY: COMPUTER SECURITY

SUBCATEGORY: CRYPTOGRAPHY

Information Technology Laboratory
National Institute of Standards and Technology
Gaithersburg, MD 20899-8900
http://dx.doi.org/10.6028/NIST.FIPS.186-4
Issued July 2013

U.S. Department of Commerce
Cameron F. Kerry, Acting Secretary
National Institute of Standards and Technology
Patrick D. Gallagher, Under Secretary of Commerce for Standards and Technology and Director

FOREWORD
The Federal Information Processing Standards Publication Series of the National Instituteof Standards and Technology (NIST) is the official series of publications relating tostandards and guidelines adopted and promulgated under the provisions of the Federal
Information Security Management Act (FISMA) of 2002.
Comments concerning FIPS publications are welcomed and should be addressed to the
Director, Information Technology Laboratory, National Institute of Standards andTechnology, 100 Bureau Drive, Stop 8900, Gaithersburg, MD 20899-8900.
Charles Romine, Director
Information Technology Laboratory

Abstract
This Standard specifies a suite of algorithms that can be used to generate a digital signature.
Digital signatures are used to detect unauthorized modifications to data and to authenticate theidentity of the signatory. In addition, the recipient of signed data can use a digital signature as
evidence in demonstrating to a third party that the signature was, in fact, generated by theclaimed signatory. This is known as non-repudiation, since the signatory cannot easily repudiate
the signature at a later time.
Key words: computer security, cryptography, digital signatures, Federal Information Processing
Standards, public key cryptography.

Federal Information Processing Standards Publication 186-4
July 2013
Announcing the
DIGITAL SIGNATURE STANDARD (DSS)
Federal Information Processing Standards Publications (FIPS PUBS) are issued by the National
Institute of Standards and Technology (NIST) after approval by the Secretary of Commerce
pursuant to Section 5131 of the Information Technology Management Reform Act of 1996
(Public Law 104-106), and the Computer Security Act of 1987 (Public Law 100-235).
1. Name of Standard: Digital Signature Standard (DSS) (FIPS 186-4).
2. Category of Standard: Computer Security. Subcategory. Cryptography.
3. Explanation: This Standard specifies algorithms for applications requiring a digital
signature, rather than a written signature. A digital signature is represented in a computer as astring of bits. A digital signature is computed using a set of rules and a set of parameters thatallow the identity of the signatory and the integrity of the data to be verified. Digital signatures
may be generated on both stored and transmitted data.
Signature generation uses a private key to generate a digital signature; signature verification uses
a public key that corresponds to, but is not the same as, the private key. Each signatory
possesses a private and public key pair. Public keys may be known by the public; private keys
are kept secret. Anyone can verify the signature by employing the signatory’s public key. Only
the user that possesses the private key can perform signature generation.
A hash function is used in the signature generation process to obtain a condensed version of thedata to be signed; the condensed version of the data is often called a message digest. The
message digest is input to the digital signature algorithm to generate the digital signature. The
hash functions to be used are specified in the Secure Hash Standard (SHS), FIPS 180. FIPS
approved digital signature algorithms shall be used with an appropriate hash function that is
specified in the SHS.
The digital signature is provided to the intended verifier along with the signed data. The
verifying entity verifies the signature by using the claimed signatory’s public key and the samehash function that was used to generate the signature. Similar procedures may be used togenerate and verify signatures for both stored and transmitted data.
4. Approving Authority: Secretary of Commerce.
i5. Maintenance Agency: Department of Commerce, National Institute of Standards andTechnology, Information Technology Laboratory, Computer Security Division.
6. Applicability: This Standard is applicable to all Federal departments and agencies for theprotection of sensitive unclassified information that is not subject to section 2315 of Title 10,
United States Code, or section 3502 (2) of Title 44, United States Code. This Standard shall be
used in designing and implementing public key-based signature systems that Federal
departments and agencies operate or that are operated for them under contract. The adoption anduse of this Standard is available to private and commercial organizations.
7. Applications: A digital signature algorithm allows an entity to authenticate the integrity ofsigned data and the identity of the signatory. The recipient of a signed message can use a digital
signature as evidence in demonstrating to a third party that the signature was, in fact, generated
by the claimed signatory. This is known as non-repudiation, since the signatory cannot easily
repudiate the signature at a later time. A digital signature algorithm is intended for use inelectronic mail, electronic funds transfer, electronic data interchange, software distribution, data
storage, and other applications that require data integrity assurance and data origin
authentication.
8. Implementations: A digital signature algorithm may be implemented in software, firmware,
hardware or any combination thereof. NIST has developed a validation program to test
implementations for conformance to the algorithms in this Standard. Information about thevalidation program is available at http://csrc.nist.gov/cryptval. Examples for each digital
signature algorithm are available at http://csrc.nist.gov/groups/ST/toolkit/examples.html.
Agencies are advised that digital signature key pairs shall not be used for other purposes.
9. Other Approved Security Functions: Digital signature implementations that comply withthis Standard shall employ cryptographic algorithms, cryptographic key generation algorithms,
and key establishment techniques that have been approved for protecting Federal government
sensitive information. Approved cryptographic algorithms and techniques include those that areeither:a. specified in a Federal Information Processing Standard (FIPS),b. adopted in a FIPS or a NIST Recommendation, orc. specified in the list of approved security functions for FIPS 140.
10. Export Control: Certain cryptographic devices and technical data regarding them aresubject to Federal export controls. Exports of cryptographic modules implementing this Standard
and technical data regarding them must comply with these Federal regulations and be licensed by
the Bureau of Industry and Security of the U.S. Department of Commerce. Information about
export regulations is available at: http://www.bis.doc.gov.
11. Patents: The algorithms in this Standard may be covered by U.S. or foreign patents.

ii

12. Implementation Schedule: This Standard becomes effective immediately upon
approval by the Secretary of Commerce. A transition strategy for validating algorithms
and cryptographic modules will be posted on NIST’s Web page at
http://csrc.nist.gov/groups/STM/cmvp/index.html under Notices. The transition planaddresses the transition by Federal agencies from modules tested and validated forcompliance to previous versions of this Standard to modules tested and validated forcompliance to FIPS 186-4 under the Cryptographic Module Validation Program. The
transition plan allows Federal agencies and vendors to make a smooth transition to FIPS
186-4.
13. Specifications: Federal Information Processing Standard (FIPS) 186-4 Digital SignatureStandard (affixed).
14. Cross Index: The following documents are referenced in this Standard. Unless a specificversion or date is indicated with the document number, the latest version of the given document
is intended as the reference.
a. FIPS PUB 140, Security Requirements for Cryptographic Modules.
b. FIPS PUB 180 Secure Hash Standard.
c. ANS X9.31-1998, Digital Signatures Using Reversible Public Key Cryptography for the
Financial Services Industry (rDSA).
d. ANS X9.62-2005, Public Key Cryptography for the Financial Services Industry: The
Elliptic Curve Digital Signature Algorithm (ECDSA).
e. ANS X9.80, Prime Number Generation, Primality Testing and Primality Certificates.
f. Public Key Cryptography Standard (PKCS) #1, RSA Encryption Standard.
g. Special Publication (SP) 800-57, Recommendation for Key Management.
h. Special Publication (SP) 800-89, Recommendation for Obtaining Assurances for Digital
Signature Applications.
i. Special Publication (SP) 800-90A, Recommendation for Random Number Generation
Using Deterministic Random Bit Generators.
j. Special Publication (SP) 800-102, Recommendation for Digital Signature Timeliness.
k. Special Publication (SP) 800-131A, Transitions: Recommendation for Transitioning theUse of Cryptographic Algorithms and Key Lengths.
l. IEEE Std. 1363-2000, Standard Specifications for Public Key Cryptography.
15. Qualifications: The security of a digital signature system is dependent on maintaining thesecrecy of the signatory’s private keys. Signatories shall, therefore, guard against the disclosureof their private keys. While it is the intent of this Standard to specify general security
requirements for generating digital signatures, conformance to this Standard does not assure thatiii

a particular implementation is secure. It is the responsibility of an implementer to ensure thatany module that implements a digital signature capability is designed and built in a securemanner.
Similarly, the use of a product containing an implementation that conforms to this Standard does
not guarantee the security of the overall system in which the product is used. The responsible
authority in each agency or department shall assure that an overall implementation provides anacceptable level of security.
Since a standard of this nature must be flexible enough to adapt to advancements andinnovations in science and technology, this Standard will be reviewed every five years in order
to assess its adequacy.
16. Waiver Procedure: The Federal Information Security Management Act (FISMA) does notallow for waivers to Federal Information Processing Standards (FIPS) that are made mandatory
by the Secretary of Commerce.
17. Where to Obtain Copies of the Standard: This publication is available by accessing
http://csrc.nist.gov/publications/. Other computer security publications are available at the same
web site.

iv

Table of Contents
1. INTRODUCTION .................................................................................................................................... 1
2. GLOSSARY OF TERMS, ACRONYMS AND MATHEMATICAL SYMBOLS ....................................... 2
2.1
2.2
2.3

TERMS AND DEFINITIONS ................................................................................................................ 2
ACRONYMS ..................................................................................................................................... 5
MATHEMATICAL SYMBOLS ................................................................................................................ 6

3. GENERAL DISCUSSION ....................................................................................................................... 9
3.1
3.2
3.3
4

INITIAL SETUP ............................................................................................................................... 11
DIGITAL SIGNATURE GENERATION .................................................................................................. 12
DIGITAL SIGNATURE VERIFICATION AND VALIDATION ....................................................................... 13

THE DIGITAL SIGNATURE ALGORITHM (DSA) ............................................................................... 15
4.1
4.2
4.3

DSA PARAMETERS ....................................................................................................................... 15
SELECTION OF PARAMETER SIZES AND HASH FUNCTIONS FOR DSA ................................................ 15
DSA DOMAIN PARAMETERS........................................................................................................... 16
4.3.1 Domain Parameter Generation ...................................................................................... 17
4.3.2

4.4

KEY PAIRS .................................................................................................................................... 17
4.4.1 DSA Key Pair Generation .............................................................................................. 17
4.4.2

4.5
4.6
4.7

Domain Parameter Management ................................................................................... 17

Key Pair Management ................................................................................................... 18

DSA PER-MESSAGE SECRET NUMBER .......................................................................................... 18
DSA SIGNATURE GENERATION ...................................................................................................... 19
DSA SIGNATURE VERIFICATION AND VALIDATION ............................................................................ 19

5. THE RSA DIGITAL SIGNATURE ALGORITHM.................................................................................. 22
5.1
5.2
5.3
5.4
5.5

RSA KEY PAIR GENERATION ......................................................................................................... 22
KEY PAIR MANAGEMENT ................................................................................................................ 23
ASSURANCES ................................................................................................................................ 23
ANS X9.31 .................................................................................................................................. 24
PKCS #1 ..................................................................................................................................... 24

6. THE ELLIPTIC CURVE DIGITAL SIGNATURE ALGORITHM (ECDSA) ............................................. 26
6.1

ECDSA DOMAIN PARAMETERS...................................................................................................... 26
6.1.1 Domain Parameter Generation ...................................................................................... 26
6.1.2

6.2

PRIVATE/PUBLIC KEYS .................................................................................................................. 28
6.2.1 Key Pair Generation ....................................................................................................... 29
6.2.2

6.3
6.4
6.5

Domain Parameter Management ................................................................................... 28

Key Pair Management ................................................................................................... 29

SECRET NUMBER GENERATION...................................................................................................... 29
ECDSA DIGITAL SIGNATURE GENERATION AND VERIFICATION ........................................................ 29
ASSURANCES ................................................................................................................................ 30

APPENDIX A: GENERATION AND VALIDATION OF FFC DOMAIN PARAMETERS ........................... 31

v

A.1

GENERATION OF THE FFC PRIMES P AND Q ................................................................................... 31
A.1.1
Generation and Validation of Probable Primes ............................................................. 31
A.1.2

A.2

Construction and Validation of the Provable Primes p and q......................................... 36

GENERATION OF THE GENERATOR G .............................................................................................. 41
A.2.1 Unverifiable Generation of the Generator g ................................................................... 41
A.2.2

Assurance of the Validity of the Generator g ................................................................. 42

A.2.3

Verifiable Canonical Generation of the Generator g ...................................................... 42

A.2.4

Validation Routine when the Canonical Generation of the Generator g Routine Was
Used ............................................................................................................................... 44

APPENDIX B: KEY PAIR GENERATION .................................................................................................. 46
B.1

FFC KEY PAIR GENERATION.......................................................................................................... 46
B.1.1 Key Pair Generation Using Extra Random Bits ............................................................. 46
B.1.2

B.2

FFC PER-MESSAGE SECRET NUMBER GENERATION ...................................................................... 48
B.2.1 Per-Message Secret Number Generation Using Extra Random Bits ............................ 49
B.2.2

B.3

B.4

Per-Message Secret Number Generation by Testing Candidates ................................. 49

IFC KEY PAIR GENERATION ........................................................................................................... 50
B.3.1 Criteria for IFC Key Pairs ............................................................................................... 50
B.3.2

Generation of Random Primes that are Provably Prime ................................................ 53

B.3.3

Generation of Random Primes that are Probably Prime ................................................ 55

B.3.4

Generation of Provable Primes with Conditions Based on Auxiliary Provable Primes .. 56

B.3.5

Generation of Probable Primes with Conditions Based on Auxiliary Provable Primes .. 58

B.3.6

Generation of Probable Primes with Conditions Based on Auxiliary Probable Primes .. 60

ECC KEY PAIR GENERATION ......................................................................................................... 61
B.4.1 Key Pair Generation Using Extra Random Bits ............................................................. 62
B.4.2

B.5

Key Pair Generation by Testing Candidates .................................................................. 47

Key Pair Generation by Testing Candidates .................................................................. 63

ECC PER-MESSAGE SECRET NUMBER GENERATION ...................................................................... 64
B.5.1 Per-Message Secret Number Generation Using Extra Random Bits ............................ 64
B.5.2

Per-Message Secret Number Generation by Testing Candidates ................................. 65

APPENDIX C: GENERATION OF OTHER QUANTITIES.......................................................................... 67
C.1
C.2

COMPUTATION OF THE INVERSE VALUE .......................................................................................... 67
CONVERSION BETWEEN BIT STRINGS AND INTEGERS ...................................................................... 68
C.2.1 Conversion of a Bit String to an Integer ......................................................................... 68
C.2.2

C.3

Conversion of an Integer to a Bit String ......................................................................... 68

PROBABILISTIC PRIMALITY TESTS ................................................................................................... 69
C.3.1 Miller-Rabin Probabilistic Primality Test ......................................................................... 71
C.3.2

Enhanced Miller-Rabin Probabilistic Primality Test ....................................................... 72vi

C.3.3 (GENERAL) LUCAS PROBABILISTIC PRIMALITY TEST ........................................................................ 74
C.4 CHECKING FOR A PERFECT SQUARE .............................................................................................. 75
C.5 JACOBI SYMBOL ALGORITHM.......................................................................................................... 76
C.6 SHAWE-TAYLOR RANDOM_PRIME ROUTINE .................................................................................... 77
C.7 TRIAL DIVISION ............................................................................................................................. 80
C.8 SIEVE PROCEDURE ....................................................................................................................... 80
C.9 COMPUTE A PROBABLE PRIME FACTOR BASED ON AUXILIARY PRIMES ............................................. 81
C.10 CONSTRUCT A PROVABLE PRIME (POSSIBLY WITH CONDITIONS), BASED ON CONTEMPORANEOUSLYCONSTRUCTED AUXILIARY PROVABLE PRIMES ................................................................................ 83
APPENDIX D: RECOMMENDED ELLIPTIC CURVES FOR FEDERAL GOVERNMENT USE ................ 87
D.1

D.2

NIST RECOMMENDED ELLIPTIC CURVES ........................................................................................ 87
D.1.1 Choices .......................................................................................................................... 87
D.1.2

Curves over Prime Fields ............................................................................................... 89

D.1.3

Curves over Binary Fields .............................................................................................. 92

IMPLEMENTATION OF MODULAR ARITHMETIC................................................................................. 101
D.2.1 Curve P-192 ................................................................................................................. 101
D.2.2

Curve P-224 ................................................................................................................. 102

D.2.3

Curve P-256 ................................................................................................................. 102

D.2.4

Curve P-384 ................................................................................................................. 103

D.2.5

Curve P-521 ................................................................................................................. 104

D.3 NORMAL BASES .......................................................................................................................... 104
D.4 SCALAR MULTIPLICATION ON KOBLITZ CURVES ............................................................................. 106
D.5 GENERATION OF PSEUDO-RANDOM CURVES (PRIME CASE) ......................................................... 109
D.6 VERIFICATION OF CURVE PSEUDO-RANDOMNESS (PRIME CASE) ................................................... 110
D.7 GENERATION OF PSEUDO-RANDOM CURVES (BINARY CASE)......................................................... 111
D.8 VERIFICATION OF CURVE PSEUDO-RANDOMNESS (BINARY CASE).................................................. 111
D.9 POLYNOMIAL BASIS TO NORMAL BASIS CONVERSION .................................................................... 112
D.10 NORMAL BASIS TO POLYNOMIAL BASIS CONVERSION .................................................................... 113
APPENDIX E: A PROOF THAT V = R IN THE DSA ................................................................................ 115
APPENDIX F: CALCULATING THE REQUIRED NUMBER OF ROUNDS OF TESTING USING THE
MILLER-RABIN PROBABILISTIC PRIMALITY TEST ...................................................................... 117
F.1
F.2
F.3

THE REQUIRED NUMBER OF ROUNDS OF THE MILLER-RABIN PRIMALITY TESTS .............................. 117
GENERATING DSA PRIMES .......................................................................................................... 118
GENERATING PRIMES FOR RSA SIGNATURES .............................................................................. 119

APPENDIX G: REFERENCES ................................................................................................................. 121

vii

Federal Information Processing Standards Publication 186-4
July 2013
Specifications for the
DIGITAL SIGNATURE STANDARD (DSS)

1.

Introduction

This Standard defines methods for digital signature generation that can be used for the protectionof binary data (commonly called a message), and for the verification and validation of thosedigital signatures. Three techniques are approved.
(1) The Digital Signature Algorithm (DSA) is specified in this Standard. The specification
includes criteria for the generation of domain parameters, for the generation of public andprivate key pairs, and for the generation and verification of digital signatures.
(2) The RSA digital signature algorithm is specified in American National Standard (ANS)
X9.31 and Public Key Cryptography Standard (PKCS) #1. FIPS 186-4 approves the useof implementations of either or both of these standards and specifies additional
requirements.
(3) The Elliptic Curve Digital Signature Algorithm (ECDSA) is specified in ANS X9.62.
FIPS 186-4 approves the use of ECDSA and specifies additional requirements.
Recommended elliptic curves for Federal Government use are provided herein.
This Standard includes requirements for obtaining the assurances necessary for valid digital
signatures. Methods for obtaining these assurances are provided in NIST Special Publication(SP) 800-89, Recommendation for Obtaining Assurances for Digital Signature Applications.

12.

Glossary of Terms, Acronyms and Mathematical Symbols

2.1

Terms and Definitions

Approved

FIPS-approved and/or NIST-recommended. An algorithm or techniquethat is either 1) specified in a FIPS or NIST Recommendation, or 2)
adopted in a FIPS or NIST Recommendation or 3) specified in a list of
NIST approved security functions.

Assurance of domain
parameter validity

Confidence that the domain parameters are arithmetically correct.

Assurance ofpossession

Confidence that an entity possesses a private key and any associatedkeying material.

Assurance of public
key validity

Confidence that the public key is arithmetically correct.

Bit string

An ordered sequence of 0’s and 1’s. The leftmost bit is the mostsignificant bit of the string. The rightmost bit is the least significant bitof the string.

Certificate

A set of data that uniquely identifies a key pair and an owner that is
authorized to use the key pair. The certificate contains the owner’s
public key and possibly other information, and is digitally signed by a
Certification Authority (i.e., a trusted party), thereby binding the
public key to the owner.

Certification Authority(CA)The entity in a Public Key Infrastructure (PKI) that is responsible for
issuing certificates and exacting compliance with a PKI policy.

Claimed signatory

From the verifier’s perspective, the claimed signatory is the entity that
purportedly generated a digital signature.

Digital signature

The result of a cryptographic transformation of data that, whenproperly implemented, provides a mechanism for verifying origin
authentication, data integrity and signatory non-repudiation.

Domain parameter seed

A string of bits that is used as input for a domain parameter generation
or validation process.

Domain parameters

Parameters used with cryptographic algorithms that are usuallycommon to a domain of users. A DSA or ECDSA cryptographic key
pair is associated with a specifc set of domain parameters.

2

Entity

An individual (person), organization, device or process. Used
interchangeably with “party”.

Equivalent process

Two processes are equivalent if, when the same values are input toeach process (either as input parameters or as values made available
during the process or both), the same output is produced.

Hash function

A function that maps a bit string of arbitrary length to a fixed length
bit string. Approved hash functions are specified in FIPS 180 and aredesigned to satisfy the following properties:
1. (One-way) It is computationally infeasible to find any input thatmaps to any new pre-specified output, and2. (Collision resistant) It is computationally infeasible to find any
two distinct inputs that map to the same output.

Hash value

See “message digest”.

Intended signatory

An entity that intends to generate digital signatures in the future.

Key

A parameter used in conjunction with a cryptographic algorithm that
determines its operation. Examples applicable to this Standard
include:1. The computation of a digital signature from data, and
2. The verification of a digital signature.

Key pair

A public key and its corresponding private key.

Message

The data that is signed. Also known as “signed data” during thesignature verification and validation process.

Message digest

The result of applying a hash function to a message. Also known as“hash value”.

Non-repudiation

A service that is used to provide assurance of the integrity and originof data in such a way that the integrity and origin can be verified andvalidated by a third party as having originated from a specific entity inpossession of the private key (i.e., the signatory).

Owner

A key pair owner is the entity that is authorized to use the private keyof a key pair.

Party

An individual (person), organization, device or process. Used
interchangeably with “entity”.

Per-message secret
number

A secret random number that is generated prior to the generation ofeach digital signature.

3

Public Key
Infrastructure (PKI)

A framework that is established to issue, maintain and revoke publickey certificates.

Prime number
generation seed

A string of random bits that is used to determine a prime number withthe required characteristics.

Private key

A cryptographic key that is used with an asymmetric (public key)
cryptographic algorithm. For digital signatures, the private key isuniquely associated with the owner and is not made public. The
private key is used to compute a digital signature that may be verified
using the corresponding public key.

Probable prime

An integer that is believed to be prime, based on a probabilistic
primality test. There should be no more than a negligible probability
that the so-called probable prime is actually composite.

Provable prime

An integer that is either constructed to be prime or is calculated to be
prime using a primality-proving algorithm.

Pseudorandom

A process or data produced by a process is said to be pseudorandom
when the outcome is deterministic, yet also effectively random as longas the internal action of the process is hidden from observation. For
cryptographic purposes, “effectively” means “within the limits of theintended security strength.”

Public key

A cryptographic key that is used with an asymmetric (public key)
cryptographic algorithm and is associated with a private key. The
public key is associated with an owner and may be made public. In thecase of digital signatures, the public key is used to verify a digital
signature that was signed using the corresponding private key.

Security strength

A number associated with the amount of work (that is, the number ofoperations) that is required to break a cryptographic algorithm orsystem. Sometimes referred to as a security level.

Shall

Used to indicate a requirement of this Standard.

ShouldUsed to indicate a strong recommendation, but not a requirement of
this Standard.

Signatory

The entity that generates a digital signature on data using a private
key.

Signature generation

The process of using a digital signature algorithm and a private key togenerate a digital signature on data.

4

Signature validation

The (mathematical) verification of the digital signature and obtaining
the appropriate assurances (e.g., public key validity, private key
possession, etc.).

Signature verification

The process of using a digital signature algorithm and a public key toverify a digital signature on data.

Signed data

The data or message upon which a digital signature has been
computed. Also, see “message”.

Subscriber

An entity that has applied for and received a certificate from a
Certificate Authority.

Trusted third party(TTP)

An entity other than the owner and verifier that is trusted by the owneror the verifier or both. Sometimes shortened to “trusted party”.

Verifier

The entity that verifies the authenticity of a digital signature using the
public key.

2.2

Acronyms

ANS

American National Standard.

CA

Certification Authority.

DSA

Digital Signature Algorithm; specified in this Standard.

ECDSA

Elliptic Curve Digital Signature Algorithm; specified in ANS X9.62.

FIPS

Federal Information Processing Standard.

NIST

National Institute of Standards and Technology.

PKCS

Public Key Cryptography Standard.

PKI

Public Key Infrastructure.

RBG

Random Bit Generator.

RSA

Algorithm developed by Rivest, Shamir and Adleman; specified in
ANS X9.31 and PKCS #1.

SHA

Secure Hash Algorithm; specified in FIPS 180.

SP

NIST Special Publication

TTP

Trusted Third Party.

5

2.3

Mathematical Symbols

a mod nThe unique remainder r, 0 ≤ r ≤ (n – 1), when integer a is divided by
the positive integer n. For example, 23 mod 7 = 2.

b ≡ a mod n

There exists an integer k such that b – a = kn; equivalently, a mod n= b mod n.

counter

The counter value that results from the domain parameter generation
process when the domain parameter seed is used to generate DSA
domain parameters.

d1. For RSA, the private signature exponent of a private key.
2. For ECDSA, the private key.

domain_parameter_seed

A seed used for the generation of domain parameters.

e

The public verification exponent of an RSA public key.

g

One of the DSA domain parameters; g is a generator of the q-order
cyclic group of GF(p)*; that is, an element of order q in themultiplicative group of GF(p).

GCD (a, b)Greatest common divisor of the integers a and b.

Hash (M)The result of a hash computation (message digest or hash value) on
message M using an approved hash function.

index

A value used in the generation of g to indicate its intended use (e.g.,
for digital signatures).

k

For DSA and ECDSA, a per-message secret number.

L

For DSA, the length of the parameter p in bits.

(L, N)

The associated pair of length parameters for a DSA key pair, where L
is the length of p, and N is the length of q.

LCM (a, b)

The least common multiple of the integers a and b.

len (a)The length of a in bits; the integer L, where 2L-1 ≤ a < 2L.

M

The message that is signed using the digital signature algorithm.

m

For ECDSA, the degree of the finite field GF2m .

N

For DSA, the length of the parameter q in bits.

6

n

1. For RSA, the modulus; the bit length of n is considered to be thekey size.
2. For ECDSA, the order of the base point of the elliptic curve; thebit length of n is considered to be the key size.

(n, d)An RSA private key, where n is the modulus, and d is the private
signature exponent.

(n, e)An RSA public key, where n is the modulus, and e is the publicverification exponent.

nlen

The length of the RSA modulus n in bits.

p

1. For DSA, one of the DSA domain parameters; a prime numberthat defines the Galois Field GF(p) and is used as a modulus in theoperations of GF(p).
2. For RSA, a prime factor of the modulus n.

q

1. For DSA, one of the DSA domain parameters; a prime factor ofp – 1.
2. For RSA, a prime factor of the modulus n.

Q

An ECDSA public key.

r

One component of a DSA or ECDSA digital signature. See thedefinition of (r, s).

(r, s)A DSA or ECDSA digital signature, where r and s are the digital
signature components.

s

One component of a DSA or ECDSA digital signature. See thedefinition of (r, s).

seedlen

The length of the domain_parameter_seed in bits.

SHAx(M)

The result when M is the input to the SHA-x hash function, where
SHA-x is specified in FIPS 180.

x

The DSA private key.

y

The DSA public key.

⊕

Bitwise logical “exclusive-or” on bit strings of the same length; forcorresponding bits of each bit string, the result is determined asfollows: 0 ⊕ 0 = 0, 0 ⊕ 1 = 1, 1 ⊕ 0 = 1, or 1 ⊕ 1 = 0.
Example:+01101 ⊕ 11010 = 10111

Addition.

7

∗Multiplication.

/

Division.

a || b

The concatenation of two strings a and b. Either a and b are both bit
strings, or both are byte strings.

a

The ceiling of a: the smallest integer that is greater than or equal toa. For example, 5 = 5, 5.3 = 6, and –2.1 = –2.

a

The floor of a; the largest integer that is less than or equal to a. For
example, 5 = 5, 5.3 = 5, and -2.1. = -3.

|a|

The absolute value of a; |a| is – a if a < 0; otherwise, it is simply a.
For example, |2| = 2, and |–2| = 2.

[a, b]The interval of integers between and including a and b. For example,[1, 4] consists of the integers 1, 2, 3 and 4.

{, a, b, …}Used to indicate optional information.

0x

The prefix to a bit string that is represented in hexadecimal characters.

8

3.

General Discussion

A digital signature is an electronic analogue of a written signature; the digital signature can beused to provide assurance that the claimed signatory signed the information. In addition, a
digital signature may be used to detect whether or not the information was modified after it was
signed (i.e., to detect the integrity of the signed data). These assurances may be obtained whether
the data was received in a transmission or retrieved from storage. A properly implemented
digital signature algorithm that meets the requirements of this Standard can provide these
services.

Signature Generation

PrivateKey

Signature Verification

Message/Data

Message/Data

Hash Function

Hash Function

Message Digest

Message Digest

Signature
Generation

PublicKey
Signature

Signature
Verification

Valid orInvalid

Figure 1: Digital Signature Processes

A digital signature algorithm includes a signature generation process and a signature verification
process. A signatory uses the generation process to generate a digital signature on data; a verifier
uses the verification process to verify the authenticity of the signature. Each signatory has apublic and private key and is the owner of that key pair. As shown in Figure 1, the private key isused in the signature generation process. The key pair owner is the only entity that is authorized
to use the private key to generate digital signatures. In order to prevent other entities fromclaiming to be the key pair owner and using the private key to generate fraudulent signatures, the9private key must remain secret. The approved digital signature algorithms are designed toprevent an adversary who does not know the signatory’s private key from generating the same
signature as the signatory on a different message. In other words, signatures are designed so thatthey cannot be forged. A number of alternative terms are used in this Standard to refer to thesignatory or key pair owner. An entity that intends to generate digital signatures in the future
may be referred to as the intended signatory. Prior to the verification of a signed message, thesignatory is referred to as the claimed signatory until such time as adequate assurance can beobtained of the actual identity of the signatory.
The public key is used in the signature verification process (see Figure 1). The public key neednot be kept secret, but its integrity must be maintained. Anyone can verify a correctly signed
message using the public key.
For both the signature generation and verification processes, the message (i.e., the signed data) isconverted to a fixed-length representation of the message by means of an approved hashfunction. Both the original message and the digital signature are made available to a verifier.
A verifier requires assurance that the public key to be used to verify a signature belongs to theentity that claims to have generated a digital signature (i.e., the claimed signatory). That is, averifier requires assurance that the signatory is the actual owner of the public/private key pairused to generate and verify a digital signature. A binding of an owner’s identity and the owner’s
public key shall be effected in order to provide this assurance.
A verifier also requires assurance that the key pair owner actually possesses the private key
associated with the public key, and that the public key is a mathematically correct key.
By obtaining these assurances, the verifier has assurance that if the digital signature can becorrectly verified using the public key, the digital signature is valid (i.e., the key pair ownerreally signed the message). Digital signature validation includes both the (mathematical)
verification of the digital signature and obtaining the appropriate assurances. The following arereasons why such assurances are required.
1. If a verifier does not obtain assurance that a signatory is the actual owner of the key pairwhose public component is used to verify a signature, the problem of forging a signatureis reduced to the problem of falsely claiming an identity. For example, anyone inpossession of a mathematically consistent key pair can sign a message and claim that thesignatory was the President of the United States. If a verifier does not require assurance
that the President is actually the owner of the public key that is used to mathematicallyverify the message’s signature, then successful signature verification provides assurance
that the message has not been altered since it was signed, but does not provide assurance
that the message came from the President (i.e., the verifier has assurance of the data’s
integrity, but source authentication is lacking).
2. If the public key used to verify a signature is not mathematically valid, the arguments
used to establish the cryptographic strength of the signature algorithm may not apply.

10

The owner may not be the only party who can generate signatures that can be verified
with that public key.
3. If a public key infrastructure cannot provide assurance to a verifier that the owner of a
key pair has demonstrated knowledge of a private key that corresponds to the owner’s
public key, then it may be possible for an unscrupulous entity to have their identity (or anassumed identity) bound to a public key that is (or has been) used by another party. The
unscrupulous entity may then claim
Intended
to be the source of certain messages
signatory
Obtainsigned by that other party. Or, itORDomain Parameters
another
may be possible that anentityDSA
unscrupulous entity has managed togenerates
andobtain ownership of a public key
ECDSA
Obtain Assurance ofthat was chosen with the sole
Domain Parameter
purpose of allowing for the
Validity
verification of a signature on a
specific message.
Technically, a key pair used by a digital
signature algorithm could also be used for
purposes other than digital signatures (e.g.,
for key establishment). However, a key pairused for digital signature generation andverification as specified in this Standard
shall not be used for any other purpose. See
SP 800-57 on Key Usage for further
information.

IntendedsignatoryORa TTP
generates

ObtainDS Key Pair

ObtainAssurance of
Public Key Validity

A number of steps are required to enable a
digital signature generation or verification
capability in accordance with this Standard.
All parties that generate digital signatures
shall perform the initial setup process as
discussed in Section 3.1. Digital signature
generation shall be performed as discussed
in Section 3.2. Digital signature verificationand validation shall be performed as
discussed in Section 3.3.

ObtainAssurance of Possessionof theDS Private Key

Optional

Register the Public Key
and Identity with a
TTP

Intended Signatory Ready for
Generating Digital Signatures

3.1

Initial Setup

Figure 2 depicts the steps that are
performed prior to generating a digital

Figure 2: Initial Setup by an Intended
Signatory11

signature by an entity intending to act as a signatory.
For the DSA and ECDSA algorithms, the intended signatory shall first obtain appropriate
domain parameters, either by generating the domain parameters itself, or by obtaining domain
parameters that another entity has generated. Having obtained the set of domain parameters, theintended signatory shall obtain assurance of the validity of those domain parameters; approved
methods for obtaining this assurance are provided in SP 800-89. Note that the RSA algorithm
does not use domain parameters.
Each intended signatory shall obtain a digital signature key pair that is generated as specified forthe appropriate digital signature algorithm, either by generating the key pair itself or byobtaining the key pair from a trusted party. The intended signatory is authorized to use the key
pair and is the owner of that key pair. Note that if a trusted party generates the key pair, thatparty needs to be trusted not to masquerade as the owner, even though the trusted party knows
the private key.
After obtaining the key pair, the intended signatory (now the key pair owner) shall obtain (1)assurance of the validity of the public key and (2) assurance that he/she actually possesses theassociated private key. Approved methods for obtaining these assurances are provided in SP
800-89.
A digital signature verifier requires assurance of the identity of the signatory. Depending on theenvironment in which digital signatures are generated and verified, the key pair owner (i.e., theintended signatory) may register the public key and establish proof of identity with a mutuallytrusted party. For example, a certification authority (CA) could sign credentials containing anowner’s public key and identity to form a certificate after being provided with proof of theowner’s identity. Systems for certifying
credentials and distributing certificates are
Generate a Message Digest
beyond the scope of this Standard. Other meansof establishing proof of identity (e.g., byproviding identity credentials along with the
public key directly to a prospective verifier) are
DSA
Obtain Additional
andInformation for theallowed.
ECDSA

3.2

Digital Signature Process

Digital Signature Generation

Figure 3 depicts the steps that are performed byan intended signatory (i.e., the entity thatgenerates a digital signature).
Prior to the generation of a digital signature, amessage digest shall be generated on theinformation to be signed using an appropriate
approved hash function.

Generate a Digital Signature

Optional

Verify the Digital Signature

Digital Signature Generation Complete

Figure 3: Digital Signature Generation
12

Depending on the digital signature algorithm to be used, additional information shall beobtained. For example, a random per-message secret number shall be obtained for DSA andECDSA.
Using the selected digital signature algorithm, the signature private key, the message digest, andany other information required by the digital signature process, a digital signature shall be
generated in accordance with this Standard.
The signatory may optionally verify the digital signature using the signature verification process
and the associated public key. This optional verification serves as a final check to detect
otherwise undetected signature generation computation errors; this verification may be prudent
when signing a high-value message, when multiple users are expected to verify the signature, orif the verifier will be verifying the signature at a much later time.
3.3

Digital Signature Verification and Validation

Figure 4 depicts the digital signature verification and validation process that are performed by averifier (e.g., the intended recipient of the signed data and associated digital signature). Note thatthe figure depicts a successful verification and validation process (i.e., no errors are detected).
In order to verify a digital signature, the verifier shall obtain the public key of the claimed
signatory, (usually) based on the claimed identity. If DSA or ECDSA has been used to generate
the digital signature, the verifier shall also obtain the domain parameters. The public key anddomain parameters may be obtained, for example, from a certificate created by a trusted party(e.g., a CA) or directly from the claimed signatory. A message digest shall be generated on thedata whose signature is to be verified (i.e., not on the received digital signature) using the samehash function that was used during the digital signature generation process. Using theappropriate digital signature algorithm, the domain parameters (if appropriate), the public key
and the newly computed message digest, the received digital signature is verified in accordancewith this Standard. If the verification process fails, no inference can be made as to whether thedata is correct, only that in using the specified public key and the specified signature format, thedigital signature cannot be verified for that data.
Before accepting the verified digital signature as valid, the verifier shall have (1) assurance ofthe signatory’s claimed identity, (2) assurance of the validity of the domain parameters (for DSA
and ECDSA), (3) assurance of the validity of the public key, and (4) assurance that the claimed
signatory actually possessed the private key that was used to generate the digital signature at thetime that the signature was generated. Methods for the verifier to obtain these assurances areprovided in SP 800-89. Note that assurance of domain parameter validity may have beenobtained during initial setup (see Section 3.1).
If the verification and assurance processes are successful, the digital signature and signed data
shall be considered valid. However, if a verification or assurance process fails, the digital
signature should be considered invalid. An organization’s policy shall govern the action to betaken for an invalid digital signature. Such policy is outside the scope of this Standard.
13

Actions

Assurances

Get the Claimed
Signatory’s Identifier

Obtain Assurance of the Claimed
Signatory’s Identity

Obtain the Domain
Parameters and Public Key

Obtain Assurance of Domain
Parameter Validity

Generate a Message Digest

Obtain Assurance of the Validityof the Owner’s Public Key

Verify the Digital Signature

Obtain Assurance that the Owner
Possesses the Private Key

Digital Signature Validation Complete

Figure 4: Digital Signature Verification and Validation

14

4

The Digital Signature Algorithm (DSA)

4.1

DSA Parameters

A DSA digital signature is computed using a set of domain parameters, a private key x, a permessage secret number k, data to be signed, and a hash function. A digital signature is verified
using the same domain parameters, a public key y that is mathematically associated with theprivate key x used to generate the digital signature, data to be verified, and the same hashfunction that was used during signature generation. These parameters are defined as follows:

4.2

pa prime modulus, where 2L–1 < p < 2L, and L is the bit length of p. Values for L areprovided in Section 4.2.

qa prime divisor of (p – 1), where 2N–1 < q < 2 N, and N is the bit length of q. Values for N
are provided in Section 4.2.

ga generator of a subgroup of order q in the multiplicative group of GF(p), such that 1 < g< p.

x

the private key that must remain secret; x is a randomly or pseudorandomly generated
integer, such that 0 < x < q, i.e., x is in the range [1, q–1].

ythe public key, where y = gx mod p.

ka secret number that is unique to each message; k is a randomly or pseudorandomly
generated integer, such that 0 < k < q, i.e., k is in the range [1, q–1].
Selection of Parameter Sizes and Hash Functions for DSA

This Standard specifies the following choices for the pair L and N (the bit lengths of p and q,
respectively):L = 1024, N = 160
L = 2048, N = 224
L = 2048, N = 256
L = 3072, N = 256
Federal Government entities shall generate digital signatures using use one or more of thesechoices.
An approved hash function, as specified in FIPS 180, shall be used during the generation ofdigital signatures. The security strength associated with the DSA digital signature process is no
greater than the minimum of the security strength of the (L, N) pair and the security strength of
the hash function that is employed. Both the security strength of the hash function used and the
security strength of the (L, N) pair shall meet or exceed the security strength required for the15

digital signature process. The security strength for each (L, N) pair and hash function is provided
in SP 800-57.
SP 800-57 provides information about the selection of the appropriate (L, N) pair in accordancewith a desired security strength for a given time period for the generation of digital signatures.
An (L, N) pair shall be chosen that protects the signed information during the entire expected
lifetime of that information. For example, if a digital signature is generated in 2009 for
information that needs to be protected for five years, and a particular (L, N) pair is invalid after
2010, then a larger (L, N) pair shall be used that remains valid for the entire period of time thatthe information needs to be protected.
It is recommended that the security strength of the (L, N) pair and the security strength of thehash function used for the generation of digital signatures be the same unless an agreement has
been made between participating entities to use a stronger hash function. When the length of theoutput of the hash function is greater than N (i.e., the bit length of q), then the leftmost N bits of
the hash function output block shall be used in any calculation using the hash function output
during the generation or verification of a digital signature. A hash function that provides a lower
security strength than the (L, N) pair ordinarily should not be used, since this would reduce thesecurity strength of the digital signature process to a level no greater than that provided by thehash function.
A Federal Government entity other than a Certification Authority (CA) should use only the first
three (L, N) pairs (i.e., the (1024, 160), (2048, 224) and (2048, 256) pairs). A CA shall use an (L,N) pair that is equal to or greater than the (L, N) pairs used by its subscribers. For example, if
subscribers are using the (2048, 224) pair, then the CA shall use either the (2048, 224), (2048,
256) or (3072, 256) pair. Possible exceptions to this rule include cross certification between
CAs, certifying keys for purposes other than digital signatures and transitioning from one key
size or algorithm to another. See SP 800-57 for further guidance.
4.3

DSA Domain Parameters

DSA requires that the private/public key pairs used for digital signature generation andverification be generated with respect to a particular set of domain parameters. These domain
parameters may be common to a group of users and may be public. A user of a set of domain
parameters (i.e., both the signatory and the verifier) shall have assurance of their validity prior tousing them (see Section 3). Although domain parameters may be public information, they shallbe managed so that the correct correspondence between a given key pair and its set of domain
parameters is maintained for all parties that use the key pair. A set of domain parameters may
remain fixed for an extended time period.
The domain parameters for DSA are the integers p, q, and g, and optionally, thedomain_parameter_seed and counter that were used to generate p and q (i.e., the full set of
domain parameters is (p, q, g {, domain_parameter_seed, counter})).

16

4.3.1 Domain Parameter Generation
Domain parameters may be generated by a trusted third party (a TTP, such as a CA) or by anentity other than a TTP. Assurance of domain parameter validity shall be obtained prior to key
pair generation, digital signature generation or digital signature verification (see Section 3).
The integers p and q shall be generated as specified in Appendix A.1. The input to thegeneration process is the selected values of L and N (see Section 4.2); the output of thegeneration process is the values for p and q, and optionally, the values of thedomain_parameter_seed and counter.
The generator g shall be generated as specified in Appendix A.2.
The security strength of a hash function used during the generation of the domain parameters
shall meet or exceed the security strength associated with the (L, N) pair. Note that this is more
restrictive than the hash function that can be used for the digital signature process (see Section
4.2).
4.3.2 Domain Parameter Management
Each digital signature key pair shall be correctly associated with one specific set of domain
parameters (e.g., by a public key certificate that identifies the domain parameters associated withthe public key). The domain parameters shall be protected from unauthorized modification until
the set is deactivated (if and when the set is no longer needed). The same domain parameters
may be used for more than one purpose (e.g., the same domain parameters may be used for both
digital signatures and key establishment). However, using different values for the generator g
reduces the risk that key pairs generated for one purpose could be accidentally used(successfully) for another purpose.
4.4

Key Pairs

Each signatory has a key pair: a private key x and a public key y that are mathematically related
to each other. The private key shall be used for only a fixed period of time (i.e., the private key
cryptoperiod) in which digital signatures may be generated; the public key may continue to be
used as long as digital signatures that were generated using the associated private key need to beverified (i.e., the public key may continue to be used beyond the cryptoperiod of the associatedprivate key). See SP 800-57 for further guidance.
4.4.1 DSA Key Pair Generation
A digital signature key pair x and y is generated for a set of domain parameters (p, q, g {,domain_parameter_seed, counter}). Methods for the generation of x and y are provided inAppendix B.1.

17

4.4.2 Key Pair Management
Guidance on the protection of key pairs is provided in SP 800-57. The secure use of digital
signatures depends on the management of an entity’s digital signature key pair as follows:
1. The validity of the domain parameters shall be assured prior to the generation of the key
pair, or the verification and validation of a digital signature (see Section 3).
2. Each key pair shall be associated with the domain parameters under which the key pair
was generated.
3. A key pair shall only be used to generate and verify signatures using the domain
parameters associated with that key pair.
4. The private key shall be used only for signature generation as specified in this Standard
and shall be kept secret; the public key shall be used only for signature verification andmay be made public.
5. An intended signatory shall have assurance of possession of the private key prior to or
concurrently with using it to generate a digital signature (see Section 3.1).
6. A private key shall be protected from unauthorized access, disclosure and modification.
7. A public key shall be protected from unauthorized modification (including substitution).
For example, public key certificates that are signed by a CA may provide such protection.
8. A verifier shall be assured of a binding between the public key, its associated domain
parameters and the key pair owner (see Section 3).
9. A verifier shall obtain public keys in a trusted manner (e.g., from a certificate signed by a
CA that the entity trusts, or directly from the intended or claimed signatory, provided thatthe entity is trusted by the verifier and can be authenticated as the source of the signed
information that is to be verified).
10. Verifiers shall be assured that the claimed signatory is the key pair owner, and that theowner possessed the private key that was used to generate the digital signature at the timethat the signature was generated (i.e., the private key that is associated with the public
key that will be used to verify the digital signature) (see Section 3.3).
11. A signatory and a verifier shall have assurance of the validity of the public key (seeSections 3.1 and 3.3).
4.5

DSA Per-Message Secret Number

A new secret random number k shall be generated prior to the generation of each digital
signature for use during the signature generation process. This secret number shall be protectedfrom unauthorized disclosure and modification.
k −1 is the multiplicative inverse of k with respect to multiplication modulo q; i.e., 0 < k −1 < q

18

and 1 = ( k −1 k) mod q. This inverse is required for the signature generation process (see Section
4.6). A technique is provided in Appendix C.1 for deriving k −1 from k.
k and k −1 may be pre-computed, since knowledge of the message to be signed is not required forthe computations. When k and k −1 are pre-computed, their confidentiality and integrity shall beprotected.
Methods for the generation of the per-message secret number are provided in Appendix B.2.
4.6

DSA Signature Generation

The intended signatory shall have assurances as specified in Section 3.1.
Let N be the bit length of q. Let min(N, outlen) denote the minimum of the positive integers N
and outlen, where outlen is the bit length of the hash function output block.
The signature of a message M consists of the pair of numbers r and s that is computed according
to the following equations:r = (gk mod p) mod q.
z = the leftmost min(N, outlen) bits of Hash(M).
s = ( k −1 (z + xr)) mod q.
When computing s, the string z obtained from Hash(M) shall be converted to an integer. The
conversion rule is provided in Appendix C.2.
Note that r may be computed whenever k, p, q and g are available, e.g., whenever the domain
parameters p, q and g are known, and k has been pre-computed (see Section 4.5), r may also be
pre-computed, since knowledge of the message to be signed is not required for the computationof r. Pre-computed k, k-1 and r values shall be protected in the same manner as the the private
key x until s has been computed (see SP 800-57).
The values of r and s shall be checked to determine if r = 0 or s = 0. If either r = 0 or s = 0, a
new value of k shall be generated, and the signature shall be recalculated. It is extremely
unlikely that r = 0 or s = 0 if signatures are generated properly.
The signature (r, s) may be transmitted along with the message to the verifier.
4.7

DSA Signature Verification and Validation

Signature verification may be performed by any party (i.e., the signatory, the intended recipientor any other party) using the signatory’s public key. A signatory may wish to verify that thecomputed signature is correct, perhaps before sending the signed message to the intended

19

recipient. The intended recipient (or any other party) verifies the signature to determine its
authenticity.
Prior to verifying the signature of a signed message, the domain parameters, and the claimed
signatory’s public key and identity shall be made available to the verifier in an authenticatedmanner. The public key may, for example, be obtained in the form of a certificate signed by atrusted entity (e.g., a CA) or in a face-to-face meeting with the public key owner.
Let M ′, r′, and s′ be the received versions of M, r, and s, respectively; let y be the public key ofthe claimed signatory; and let N be the bit length of q. Also, let min(N, outlen) denote theminimum of the positive integers N and outlen, where outlen is the bit length of the hashfunction output block.
The signature verification process is as follows:
1. The verifier shall check that 0 < r′ < q and 0 < s′ < q; if either condition is violated, thesignature shall be rejected as invalid.
2. If the two conditions in step 1 are satisfied, the verifier computes the following:w = (s′)–1 mod q.
z = the leftmost min(N, outlen) bits of Hash(M′ ).
u1 = (zw) mod q.
u2 = ((r′)w) mod q.
v = (((g)u1 (y)u2) mod p) mod q.
A technique is provided in Appendix C.1 for deriving (s′ )–1 (i.e., the multiplicative
inverse of s′ mod q).
The string z obtained from Hash(M′) shall be converted to an integer. The conversion
rule is provided in Appendix C.2.
3. If v = r′, then the signature is verified. For a proof that v = r′ when M′ = M, r′ = r, and s′= s, see Appendix E.
4. If v does not equal r′, then the message or the signature may have been modified, there
may have been an error in the signatory’s generation process, or an imposter (who did notknow the private key associated with the public key of the claimed signatory) may haveattempted to forge the signature. The signature shall be considered invalid. No inference
can be made as to whether the data is valid, only that when using the public key to verify
the signature, the signature is incorrect for that data.
5. Prior to accepting the signature as valid, the verifier shall have assurances as specified in
Section 3.3.

20

An organization’s policy may govern the action to be taken for invalid digital signatures. Suchpolicy is outside the scope of this Standard. Guidance about determining the timeliness ofdigitally signed messages is addressed in SP 800-102, Recommendation for Digital Signature
Timeliness.

21

5.

The RSA Digital Signature Algorithm

The use of the RSA algorithm for digital signature generation and verification is specified in
American National Standard (ANS) X9.31 and Public Key Cryptography Standard (PKCS) #1.
While each of these standards uses the RSA algorithm, the format of the ANS X9.31 and PKCS
#1 data on which the digital signature is generated differs in details that make the algorithms
non-interchangeable.
5.1

RSA Key Pair Generation

An RSA digital signature key pair consists of an RSA private key, which is used to compute a
digital signature, and an RSA public key, which is used to verify a digital signature. An RSA key
pair used for digital signatures shall only be used for one digital signature scheme (e.g., ANS
X9.31, RSASSA-PKCS1 v1.5 or RSASSA-PSS; see Sections 5.4 and 5.5). In addition, an RSA
digital signature key pair shall not be used for other purposes (e.g., key establishment).
An RSA public key consists of a modulus n, which is the product of two positive prime integers
p and q (i.e., n = pq), and a public key exponent e. Thus, the RSA public key is the pair of values(n, e) and is used to verify digital signatures. The size of an RSA key pair is commonly
considered to be the length of the modulus n in bits (nlen).
The corresponding RSA private key consists of the same modulus n and a private key exponent dthat depends on n and the public key exponent e. Thus, the RSA private key is the pair of values(n, d) and is used to generate digital signatures. Note that an alternative method for representing(n, d) using the Chinese Remainder Theorem (CRT) is allowed.
In order to provide security for the digital signature process, the two integers p and q, and theprivate key exponent d shall be kept secret. The modulus n and the public key exponent e may
be made known to anyone. Guidance on the protection of these values is provided in SP 800-57.
This Standard specifies three choices for the length of the modulus (i.e., nlen): 1024, 2048 and3072 bits. Federal Government entities shall generate digital signatures using one or more ofthese choices.
An approved hash function, as specified in FIPS 180, shall be used during the generation of key
pairs and digital signatures. When used during the generation of an RSA key pair (as specified inthis Standard), the length in bits of the hash function output block shall meet or exceed the
security strength associated with the bit length of the modulus n (see SP 800-57).
The security strength associated with the RSA digital signature process is no greater than theminimum of the security strength associated with the bit length of the modulus and the security
strength of the hash function that is employed. The security strength for each modulus length andhash function used during the digital signature process is provided in SP 800-57. Both thesecurity strength of the hash function used and the security strength associated with the bit length22

of the modulus n shall meet or exceed the security strength required for the digital signatureprocess.
It is recommended that the security strength of the modulus and the security strength of the hashfunction be the same unless an agreement has been made between participating entities to use astronger hash function. A hash function that provides a lower security strength than the security
strength associated with the bit length of the modulus ordinarily should not be used, since thiswould reduce the security strength of the digital signature process to a level no greater than thatprovided by the hash function.
Federal Government entities other than CAs should use only the first two choices (i.e., nlen =1024 or 2048) during the timeframes indicated in SP 800-57. A CA should use a modulus whose
length nlen is equal to or greater than the moduli used by its subscribers. For example, if thesubscribers are using nlen = 2048, then the CA should use nlen ≥ 2048. SP 800-57 provides
further information about the selection of the bit length of n. Possible exceptions to this rule
include cross certification between CAs, certifying keys for purposes other than digital
signatures and transitioning from one key size or algorithm to another.
Criteria for the generation of RSA key pairs are provided in Appendix B.3.1.
When RSA parameters are randomly generated (i.e., the primes p and q, and optionally, thepublic key exponent e), they shall be generated using an approved random bit generator. The(pseudo) random bits produced by the random bit generator shall be used as seeds for generating
RSA parameters (e.g., the (pseudo) random number is used as a prime number generation seed).
Prime number generation seeds shall be kept secret or destroyed when the modulus n iscomputed. If any prime number generation seed is retained (e.g., to regenerate the RSA modulusn, or as evidence that the generated prime factors p and q were generated in compliance with thisStandard, then the seed shall be kept secret and shall be protected. The strength of thisprotection shall be (at least) equivalent to the protection required for the associated private key.
5.2

Key Pair Management

The secure use of digital signatures depends on the management of an entity’s digital signature
key pair. Key pair management requirements for RSA are provided in Section 4.4.2,
requirements 4 – 11. Note that the first three requirements in Section 4.4.2, which address therelationship between domain parameters and key pairs, are not applicable to RSA.
5.3

Assurances

The intended signatory shall have assurances as specified in Section 3.1. Prior to accepting a digital
signature as valid, the verifier shall have assurances as specified in Section 3.3.

23

5.4

ANS X9.31

ANS X9.31, Digital Signatures Using Reversible Public Key Cryptography for the Financial
Services Industry (rDSA), was developed for the American National Standards Institute by the
Accredited Standards Committee on Financial Services, X9. See http://www.x9.org for
information about obtaining copies of ANS X9.31 and any associated errata. The following
discussions are based on the version of ANS X9.31 that was approved in 1998.
Methods for the generation of the private prime factors p and q are provided in Appendix B.3.
In ANS X9.31, the length of the modulus n is allowed in increments of 256 bits beyond aminimum of 1024 bits. Implementations claiming conformance to FIPS 186-4 shall include one
or more of the modulus sizes specified in Section 5.1.
Two methods for the generation of digital signatures are included in ANS X9.31. When the
public signature verification exponent e is odd, the digital signature algorithm is commonly
known as RSA; when the public signature verification exponent e is even, the digital signaturealgorithm is commonly known as Rabin-Williams. This Standard (i.e., FIPS 186-4) adopts theuse of RSA, but does not adopt the use of Rabin-Williams.
During signature verification, the extraction of the hash value H(M)′ from the data structure IR′
shall be accomplished by either:•

Selecting the hashlen bytes of the data structure IR′ that immediately precedes the two
bytes of trailer information, where hashlen is the length in bytes of the hash functionused, regardless of the length of the padding, or•

If the hash value H(M)′ is selected by its location with respect to the last byte of padding(i.e., 0xBA), including a check that the hash value is followed by only two bytes
containing the expected trailer value.

ANS X9.31 contains an annex on random number generation. However, implementations of
ANS X9.31 shall use approved random number generation methods.
Annexes in ANS X9.31 provide informative discussions of security and implementationconsiderations.
5.5

PKCS #1

Public-Key Cryptography Standard (PKCS) #1, RSA Cryptography Standard, defines
mechanisms for encrypting and signing data using the RSA algorithm. PKCS #1 v2.1 specifies
two digital signature processes and corresponding formats: RSASSA-PKCS1-v1.5 and
RSASSA-PSS. Both signature schemes are approved for use, but additional constraints are
imposed beyond those specified in PKCS #1 v2.1.
(a) Implementations that generate RSA key pairs shall use the criteria and methods inAppendix B.3 to generate those key pairs,

24

(b) Only approved hash functions shall be used.
(c) Only two prime factors p and q shall be used to form the modulus n.
(d) Random numbers shall be generated using an approved random bit generator.
(e) For RSASSA-PSS:•

If nlen = 1024 bits (i.e., 128 bytes), and the output length of the approved hashfunction output block is 512 bits (i.e., 64 bytes), then the length (in bytes) of thesalt (sLen) shall satisfy 0 ≤ sLen ≤ hLen – 2,•

Otherwise, the length (in bytes) of the salt (sLen) shall satisfy 0 ≤ sLen ≤ hLen,

where hLen is the length of the hash function output block (in bytes).
(f) For RSASSA-PKCS-v1.5, when the hash value is recovered from the encoded message
EM during the verification of the digital signature 1, the extraction of the hash value shallbe accomplished by either:•

Selecting the rightmost (least significant) bits of EM, based on the size of the hashfunction used, regardless of the length of the padding, or•

If the hash value is selected by its location with respect to the last byte of padding,including a check that the hash value is located in the rightmost (least significant)bytes of EM (i.e., no other information follows the hash value in the encoded
message).

Note: PKCS #1 was initially developed by RSA Laboratories in 1991 and has been revised as
multiple versions. At the time of the approval of FIPS 186-4, three versions of PKSC #1 wereavailable: version 1.5, version 2.0 and version 2.1. This Standard references only version 2.1.

1

PKCS #1, v2.1 provides two methods for comparing the hash values: by comparing the encoded messages EM andEM′, and by extracting the hash value from the decoding of the encoded message (see the Note in PKCS #1, v2.1).
Step (f) above applies to the latter case.

25

6. The Elliptic Curve Digital Signature Algorithm (ECDSA)ANS X9.62, Public Key Cryptography for the Financial Services Industry: The Elliptic Curve
Digital Signature Standard (ECDSA), was developed for the American National Standards
Institute by the Accredited Standards Committee on Financial Services, X9. Information aboutobtaining copies of ANS X9.62 is available at http://www.x9.org. The following discussions are
based on the version of ANS X9.62 that was approved in 2005. This version of ANS X9.62 shallbe used, subject to the transition period referenced in the implementation schedule of thisStandard.
ANS X9.62 defines methods for digital signature generation and verification using the Elliptic
Curve Digital Signature Algorithm (ECDSA). Specifications for the generation of the domain
parameters used during the generation and verification of digital signatures are also included in
ANS X9.62. ECDSA is the elliptic curve analog of DSA. ECDSA keys shall not be used for any
other purpose (e.g., key establishment).
6.1

ECDSA Domain Parameters

ECDSA requires that the private/public key pairs used for digital signature generation andverification be generated with respect to a particular set of domain parameters. These domain
parameters may be common to a group of users and may be public. Domain parameters may
remain fixed for an extended time period.
Domain parameters for ECDSA are of the form (q, FR, a, b {, domain_parameter_seed}, G, n,h), where q is the field size; FR is an indication of the basis used; a and b are two field elementsthat define the equation of the curve; domain_parameter_seed is the domain parameter seed andis an optional bit string that is present if the elliptic curve was randomly generated in a verifiablefashion, G is a base point of prime order on the curve (i.e., G = (x G, y G )), n is the order of thepoint G, and h is the cofactor (which is equal to the order of the curve divided by n).
6.1.1 Domain Parameter Generation
This Standard specifies five ranges for n (see Table 1). For each range, a maximum cofactor sizeis also specified. Note that the specification of a cofactor h in a set of domain parameters is
optional in ANS X9.62, whereas implementations conforming to this Standard (i.e., FIPS 186-4)
shall specify the cofactor h in the set of domain parameters. Table 1 provides the maximum sizes
for the cofactor h.

26

Table 1: ECDSA Security Parameters
Bit length of n

Maximum
Cofactor (h)160 - 223210

224 - 255214

256 - 383

216

384 - 511

224

≥ 512

232

ECDSA is defined for two arithmetic fields: the finite field GFp and the finite field GF2m . For
the field GF p , p is required to be an odd prime.
NIST Recommended curves are provided in Appendix D of this Standard (i.e., FIPS 186-4).
Three types of curves are provided:
1. Curves over prime fields, which are identified as P-xxx,
2. Curves over Binary fields, which are identified as B-xxx, and3. Koblitz curves, which are identified as K-xxx,
where xxx indicates the bit length of the field size.
Alternatively, ECDSA domain parameters may be generated as specified in ANS X9.62; when
ECDSA domain parameters are generated (i.e., the NIST Recommended curves are not used),
the value of G should be generated canonically (verifiably random). An approved hash function,as specified in FIPS 180, shall be used during the generation of ECDSA domain parameters.
When generating these domain parameters, the security strength of a hash function used shallmeet or exceed the security strength associated with the bit length of n (see footnote 2 below).
An approved hash function, as specified in FIPS 180, is required during the generation of domain
parameters. The security strength of the hash function used shall meet or exceed the security
strength associated with the bit length of n. The security strengths for the ranges of n and thehash functions are provided in SP 800-57. It is recommended that the security strength
associated with the bit length of n and the security strength of the hash function be the same

2

The NIST Recommended curves were generated prior to the formulation of this guidance and using SHA-1, which
was the only approved hash function available at that time. Since SHA-1 was considered secure at the time ofgeneration, the curves were made public, and SHA-1 will only be used to validate those curves, the NIST
Recommended curves are still considered secure and appropriate for Federal government use.

27

unless an agreement has been made between participating entities to use a stronger hashfunction; a hash function that provides a lower security strength than is associated with the bit
length of n shall not be used. If the length of the output of the hash function is greater than the
bit length of n, then the leftmost n bits of the hash function output block shall be used in any
calculation using the hash function output during the generation or verification of a digital
signature.
Normally, a CA should use a bit length of n whose assessed security strength is equal to or
greater than the assessed security strength associated with the bit length of n used by its
subscribers. For example, if subscribers are using a bit length of n with an assessed security
strength of 112 bits, then CAs should use a bit length of n whose assessed security strength isequal to or greater than 112 bits. SP 800-57 provides further information about the selection of abit length of n. Possible exceptions to this rule include cross certification between CAs,
certifying keys for purposes other than digital signatures and transitioning from one key size or
algorithm to another. However, these exceptions require further analysis.
6.1.2 Domain Parameter Management
Each ECDSA key pair shall be correctly associated with one specific set of domain parameters(e.g., by a public key certificate that identifies the domain parameters associated with the publickey). The domain parameters shall be protected from unauthorized modification until the set isdeactivated (if and when the set is no longer needed). The same domain parameters may be used
for more than one purpose (e.g., the same domain parameters may be used for both digital
signatures and key establishment). However, using different domain parameters reduces the riskthat key pairs generated for one purpose could be accidentally used (successfully) for anotherpurpose.
6.2

Private/Public Keys

An ECDSA key pair consists of a private key d and a public key Q that is associated with a
specific set of ECDSA domain parameters; d, Q and the domain parameters are mathematicallyrelated to each other. The private key is normally used for a period of time (i.e., thecryptoperiod); the public key may continue to be used as long as digital signatures that have been
generated using the associated private key need to be verified (i.e., the public key may continue
to be used beyond the cryptoperiod of the associated private key). See SP 800-57 for further
guidance.
ECDSA keys shall only be used for the generation and verification of ECDSA digital signatures.

28

6.2.1 Key Pair Generation
A digital signature key pair d and Q is generated for a set of domain parameters (q, FR, a, b {,domain_parameter_seed}, G, n, h). Methods for the generation of d and Q are provided inAppendix B.4.
6.2.2 Key Pair Management
The secure use of digital signatures depends on the management of an entity’s digital signature
key pair as specified in Section 4.4.2.
6.3

Secret Number Generation

A new secret random number k shall be generated prior to the generation of each digital
signature for use during the signature generation process. This secret number shall be protectedfrom unauthorized disclosure and modification. Methods for the generation of the per-message
secret number are provided in Appendix B.5.

k −1 is the multiplicative inverse of k with respect to multiplication modulo n; i.e., 0 < k −1 < nand 1 = ( k −1 k) mod n. This inverse is required for the signature generation process. A technique
is provided in Appendix C.1 for deriving k −1 from k.
k and k −1 may be pre-computed, since knowledge of the message to be signed is not required forthe computations. When k and k −1 are pre-computed, their confidentiality and integrity shall beprotected.
6.4

ECDSA Digital Signature Generation and Verification

An ECDSA digital signature (r, s) shall be generated as specified in ANS X9.62, using:
1. Domain parameters that are generated in accordance with Section 6.1.1,
2. A private key that is generated as specified in Section 6.2.1,
3. A per-message secret number that is generated as specified in Section 6.3,
4. An approved hash function as discussed below, and
5. An approved random bit generator.
An ECDSA digital signature shall be verified as specified in ANS X9.62, using the same domain
parameters and hash function that were used during signature generation.
An approved hash function, as specified in FIPS 180, shall be used during the generation ofdigital signatures. The security strength associated with the ECDSA digital signature process is
no greater than the minimum of the security strength associated with the bit length of n and the
security strength of the hash function that is employed. Both the security strength of the hash29

function used and the security strength associated with the bit length of n shall meet or exceed
the security strength required for the digital signature process. The security strengths for theranges of the bit lengths of n and for each hash function is provided in SP 800-57.
It is recommended that the security strength associated with the bit length of n and the security
strength of the hash function be the same unless an agreement has been made between
participating entities to use a stronger hash function. When the length of the output of the hashfunction is greater than the bit length of n, then the leftmost n bits of the hash function output
block shall be used in any calculation using the hash function output during the generation orverification of a digital signature. A hash function that provides a lower security strength than
the security strength associated with the bit length of n ordinarily should not be used, since thiswould reduce the security strength of the digital signature process to a level no greater than thatprovided by the hash function.
6.5

Assurances

The intended signatory shall have assurances as specified in Section 3.1. Prior to accepting asignature as valid, the verifier shall have assurances as specified in Section 3.3.

30

APPENDIX A: Generation and Validation of FFC Domain Parameters
Finite field cryptography (FFC) is a method of implementing discrete logarithm cryptographyusing finite field mathematics. DSA, as specified in this Standard, is an example of FFC. The
Diffie-Hellman and MQV key establishment algorithms specified in SP 800-56A can also be
implemented as FFC.
The domain parameters for FFC consist of the set of values (p, q, g {, domain_parameter_seed,counter}). This appendix specifies techniques for the generation of the FFC domain parametersp, q and g and performing an explicit domain parameter validation. During the generation
process, the values for domain_parameter_seed and counter are obtained.
A.1

Generation of the FFC Primes p and q

This section provides methods for generating the primes p and q that fulfill the criteria specified
in Sections 4.1 and 4.2. One of these methods shall be used when generating these primes. A
method is provided in Appendix A.1.1 to generate random candidate integers and then test themfor primality using a probabilistic algorithm. A second method is provided in Appendix A.1.2
that constructs integers from smaller integers so that the constructed integer is guaranteed to be
prime.
During the generation, validation and testing processes, conversions between bit strings andintegers are required. Appendix C.2 provides methods for these conversions.
A.1.1

Generation and Validation of Probable Primes

Previous versions of this Standard contained a method for the generation of the domain
parameters p and q using SHA-1 and probabilistic methods. This method is no longer approvedfor domain parameter generation; however, the validation process for this method is provided inAppendix A.1.1.1 to validate previously generated domain parameters.
A method for the generation and validation of the primes p and q using probabilistic methods isprovided in Appendix A.1.1.2 and is based on the use of an approved hash function; this method
shall be used for generating probable primes. The validation process for this method is provided
in Appendix A.1.1.3.
The probabilistic methods use a hash function and an arbitrary seed (domain_parameter_seed).
Arbitrary seeds could be anything, e.g., a user’s favorite number or a random or pseudorandom
number output by an approved random number generator. The domain_parameter_seed
determines a sequence of candidates for p and q in the required intervals that are then tested forprimality using a probabilistic primality test (see Appendix C.3). The test determines that thecandidate is either not a prime (i.e., it is a composite integer) or is “probably a prime” (i.e., there
is a very small probability that a composite integer will be declared to be a prime). p and q shallbe the first candidate set that passes the primality tests. Note that the domain_parameter_seed31shall be unique for every unique set of domain parameters that are generated using the same
method.
A.1.1.1 Validation of the Probable Primes p and q that were Generated Using SHA-1 asSpecified in Prior Versions of this Standard
This prime validation algorithm is used to validate that the primes p and q that were generated by
the prime generation algorithm specified in previous versions of this Standard. The algorithm
requires the values of p, q, domain_parameter_seed and counter, which were output from theprime generation algorithm.
Let SHA1( ) be the SHA-1 hash function specified in FIPS 180. The following process or its
equivalent shall be used to validate p and q for this method.
Input:
1. p, qThe generated primes p and q.

2. domain_parameter_seed A seed that was used to generate p and q.
3. counterA count value that was determined during generation.

Output:
1. status

The status returned from the validation procedure, where
status is either VALID or INVALID.

Process:
1. If (len (p) ≠ 1024) or (len (q) ≠ 160), then return INVALID.
2. If (counter > 4095), then return INVALID.
3. seedlen = len (domain_parameter_seed).
4. If (seedlen < 160), then return INVALID.
5. computed_q = SHA1(domain_parameter_seed) ⊕ SHA1((domain_parameter_seed +1) mod 2seedlen).
6. Set the first and last bits of computed_q equal to 1 (i.e., the 159th and 0th bits).
7. Test whether or not computed_q is prime as specified in Appendix C.3. If(computed_q ≠ q) or (computed_q is not prime), then return INVALID.
8. offset = 2.
9. For i = 0 to counter do
9.1

For j = 0 to 6 do
V j = SHA1((domain_parameter_seed + offset + j) mod 2seedlen).

9.2

W = V 0 + (V 1 ∗ 2160) + (V 2 ∗ 2320) + (V 3 ∗ 2480) + (V 4 ∗ 2640) + (V 5 ∗ 2800) +32

((V 6 mod 263) ∗ 2960).
Comment: 0 ≤ W < 2L–1.

9.3

X = W + 21023.

9.4

c = X mod 2q.

9.5

computed_p = X – (c – 1). Comment: computed_p ≡ 1 (mod 2q).

9.6

If (computed_p < 21023), then go to step 9.8.

9.7

Test whether or not computed_p is prime as specified in Appendix C.3. Ifcomputed_p is determined to be prime, then go to step 10.

9.8

offset = offset + 7.

10. If ((i ≠ counter) or (computed_p ≠ p) or (computed_p is not prime)), then return
INVALID.
11. Return VALID.
A.1.1.2 Generation of the Probable Primes p and q Using an Approved Hash Function
This method uses an approved hash function and may be used for the generation of the primes p
and q for any application (e.g., digital signatures or key establishment). The security strength of
the hash function shall be equal to or greater than the security strength associated with the (L, N)
pair.
An arbitrary domain_parameter_seed of seedlen bits is also used, where seedlen shall be equal
to or greater than N.
The generation process returns a set of integers p and q that have a very high probability of being
prime. For another entity to validate that the primes were generated correctly using the validation
process in Appendix A.1.1.3, the value of the domain_parameter_seed and the counter used togenerate the primes must also be returned and made available to the validating entity; thedomain_parameter_seed and counter need not be kept secret. Let Hash( ) be the selected hashfunction, and let outlen be the bit length of the output block, where outlen shall be equal to or
greater than N.
The following process or its equivalent shall be used to generate p and q for this method.
Input:
1. L

The desired length of the prime p (in bits).

2. N

The desired length of the prime q (in bits).

3. seedlen

The desired length of the domain parameter seed; seedlen shall beequal to or greater than N.

Output:
1. status

The status returned from the generation procedure, where status is33either VALID or INVALID. If INVALID is returned as the status,
either no values for the other output parameters shall be returned, orinvalid values shall be returned (e.g., zeros or Null strings).
2. p, qThe generated primes p and q.

3. domain_parameter_seed(Optional) A seed that was used to generate p and q.
4. counter(Optional) A count value that was determined during generation.

Process:
1. Check that the (L, N) pair is in the list of acceptable (L, N pairs) (see Section 4.2). If
the pair is not in the list, then return INVALID.
2. If (seedlen < N), then return INVALID.
3. n = L ⁄ outlen – 1.
4. b = L – 1 – (n ∗ outlen).
5. Get an arbitrary sequence of seedlen bits as the domain_parameter_seed.
6. U = Hash (domain_parameter_seed) mod 2N–1.
7. q = 2N–1 + U + 1 – ( U mod 2).
8. Test whether or not q is prime as specified in Appendix C.3.
9. If q is not a prime, then go to step 5.
10. offset = 1.
11. For counter = 0 to (4L – 1) do
11.1

For j = 0 to n do
V j = Hash ((domain_parameter_seed + offset + j) mod 2seedlen).

11.2

W = V 0 + (V 1 ∗ 2outlen) + … + (V n–1 ∗ 2(n–1) ∗ outlen) + ((V n mod 2b) ∗ 2n ∗ outlen).

11.3

X = W + 2L–1.

11.4

c = X mod 2q.

11.5

p = X – (c – 1).
L–1

Comment: 0 ≤ W < 2L–1; hence, 2L–1 ≤ X < 2L.
Comment: p ≡ 1 (mod 2q).

11.6

If (p < 2), then go to step 11.9.

11.7

Test whether or not p is prime as specified in Appendix C.3.

11.8

If p is determined to be prime, then return VALID and the values of p, q and(optionally) the values of domain_parameter_seed and counter.
34

11.9

offset = offset + n + 1.

Comment: Increment offset; then, as part ofthe loop in step 11, increment counter; if
counter < 4L, repeat steps 11.1 through 11.8.

12. Go to step 5.
A.1.1.3

Validation of the Probable Primes p and q that were Generated Using an
Approved Hash Function

This prime validation algorithm is used to validate that the integers p and q were generated by
the prime generation algorithm given in Appendix A.1.1.2. The validation algorithm requires thevalues of p, q, domain_parameter_seed and counter, which were output from the prime
generation algorithm. Let Hash( ) be the hash function used to generate p and q, and let outlen
be its output block length.
The following process or its equivalent shall be used to validate p and q for this method.
Input:
1. p, qThe generated primes p and q.

3. domain_parameter_seed The domain parameter seed that was used to generate p andq.
4. counterA count value that was determined during generation.

Output:
1. status

The status returned from the validation procedure, where
status is either VALID or INVALID.

Process:
1. L = len (p).
2. N = len (q).
3. Check that the (L, N) pair is in the list of acceptable (L, N) pairs (see Section 4.2). If thepair is not in the list, return INVALID.
4. If (counter > (4L – 1)), then return INVALID.
5. seedlen = len (domain_parameter_seed).
6. If (seedlen < N), then return INVALID.
7. U = Hash(domain_parameter_seed) mod 2N–1.
8. computed_q = 2N–1 + U + 1 – ( U mod 2).
9. Test whether or not computed_q is prime as specified in Appendix C.3. If (computed_q ≠q) or (computed_q is not prime), then return INVALID.

35

10. n = L ⁄ outlen – 1.
11. b = L – 1 – (n ∗ outlen).
12. offset = 1.
13. For i = 0 to counter do
13.1 For j = 0 to n do
V j = Hash((domain_parameter_seed + offset + j) mod 2seedlen).
13.2 W = V 0 + (V 1 ∗ 2outlen) + … + (V n–1 ∗ 2(n–1) ∗ outlen) + ((V n mod 2b) ∗ 2n ∗ outlen).
13.3 X = W + 2L–1.
13.4 c = X mod 2q.
13.5 computed_p = X – (c – 1).
13.6 If (computed_p < 2L–1), then go to step 13.9
13.7 Test whether or not computed_p is prime as specified in Appendix C.3.
13.8 If computed_p is determined to be a prime, then go to step 14.
13.9 offset = offset + n + 1.
14. If ((i ≠ counter) or (computed_p ≠ p) or (computed_p is not a prime)), then return
INVALID.
15. Return VALID.
A.1.2 Construction and Validation of the Provable Primes p and q
Primes can be generated so that they are guaranteed to be prime. The following algorithm forgenerating p and q uses an approved hash function and an arbitrary seed (firstseed) to construct p
and q in the required intervals. The security strength of the hash function shall be equal to or
greater than the security strength associated with the (L, N) pair.
Arbitrary seeds can be anything, e.g., a user’s favorite number or a random or pseudorandom
number that is output from a random number generator. Note that the firstseed must be unique to
produce a unique set of domain parameters. Candidate primes are tested for primality using adeterministic primality test that proves whether or not the candidate is prime. The resulting p andq are guaranteed to be primes.
A.1.2.1 Construction of the Primes p and q Using the Shawe-Taylor Algorithm
For each set of domain parameters generated, an arbitrary initial seed (firstseed) of at least
seedlen bits shall be determined, where seedlen shall be ≥ N.
The generation process returns a set of integers p and q that are guaranteed to be prime. For36

another entity to validate that the primes were generated correctly (using the validation process
in Appendix A.1.2.2), the value of the firstseed, the two computed seeds (pseed and qseed) andthe counters used to generate the primes (pgen_counter and qgen_counter) must be madeavailable to the validating entity; the seeds and the counters need not be kept secret. The domain
parameters for DSA are identified in Section 4.3 as (p, q, g {, domain_parameter_seed,counter}). When using the Shawe-Taylor algorithm for generating p and q, thedomain_parameter_seed consists of three seed values (firstseed, pseed, and qseed), and thecounter consists of the pair of counter values (pgen_counter and qgen_counter).
Let Hash( ) be the selected hash function (see Appendix A.1.2), and let outlen be the bit lengthof the output block of that hash function.
A.1.2.1.1 Get the First Seed
The following process or its equivalent shall be used to generate firstseed for this constructive
method.
Input:
1. N

The length of q in bits.

2. seedlen

The length of firstseed, where seedlen ≥ N.

Output:
1. status

The status returned from the generation procedure, where status iseither SUCCESS or FAILURE. If FAILURE is returned, theneither no firstseed value shall be provided or an invalid value shallbe returned.

2. firstseed

The first seed generated.

Process:
1. firstseed = 0.
2. Check that N is in the list of acceptable (L, N) pairs (see Section 4.2). If not, thenreturn FAILURE.
3. If (seedlen < N), then return FAILURE.
4. While firstseed < 2N–1.
Get an arbitrary sequence of seedlen bits as firstseed.
5. Return SUCCESS and the value of firstseed.
Note: This routine could be incorporated into the beginning of the constructive prime generation
procedure in Appendix A.1.2.1.2. However, this was not done in this specification so that the37

validation process in Appendix A.1.2.2 could also call the constructive prime generation
procedure and provide the value of firstseed as input.
A.1.2.1.2 Constructive Prime Generation
The following process or its equivalent shall be used to generate p and q for this constructive
method.
Input:
1. L

The requested length for p (in bits).

2. N

The requested length for q (in bits).

3. firstseed

The first seed to be used. This was obtained as specified inAppendix A.1.2.1.1.

Output:
1. status

The status returned from the generation procedure, where status iseither SUCCESS or FAILURE. If FAILURE is returned, theneither no other values shall be returned, or invalid values shall be
returned.

2. p, q

The requested primes.

3. pseed, qseed(Optional) Computed seed values that were used to generate p andq. The entire seed for the generation of p and q consists offirstseed, pseed and qseed.

4. pgen_counter, qgen_counter(Optional) The count values that were determined during generation.
Process:
1. Check that the (L, N) pair is in the list of acceptable (L, N) pairs (see Section 4.2). If
the pair is not in the list, return FAILURE.
Comment: Use the Shawe-Taylor random
prime routine in Appendix C.6 to generaterandom primes.
2. Using N as the length and firstseed as the input_seed, use the random prime
generation routine in Appendix C.6 to obtain q, qseed and qgen_counter. If
FAILURE is returned, then return FAILURE.
3. Using L / 2 + 1 as the length and qseed as the input_seed, use the random prime
routine in Appendix C.6 to obtain p 0 , pseed, and pgen_counter. If FAILURE isreturned, then return FAILURE.

38

4. iterations = L / outlen –1.
5. old_counter = pgen_counter.
Comment: Generate a (pseudo) random x in
the interval [2L−1, 2L].
6. x = 0.
7. For i = 0 to iterations do
x = x + (Hash(pseed + i) ∗ 2 i * outlen).
8. pseed = pseed + iterations + 1.
9. x = 2L−1 + (x mod 2L–1).
Comment: Generate p, a candidate for theprime, in the interval [2L−1, 2L].
10. t = x / (2q p 0 ).
11. If (2tq p 0 + 1) > 2L, then t = 2L−1 / (2q p 0 ).
12. p = 2tq p 0 + 1.
13. pgen_counter = pgen_counter + 1.
Comment: Test p for primality; choose an
integer a in the interval [2, p–2].
14. a = 0
15. For i = 0 to iterations doa = a + (Hash(pseed + i) ∗ 2 i * outlen).
16. pseed = pseed + iterations + 1.
17. a = 2 + (a mod (p–3)).
18. z = a2tq mod p.
19. If ((1 = GCD(z–1, p) ) and ( 1 = z p0 mod p )), then return SUCCESS and the valuesof p, q and (optionally) pseed, qseed, pgen_counter, and qgen_counter.
20. If (pgen_counter > (4L + old_counter)), then return FAILURE.
21. t = t + 1.
22. Go to step 11.

39

A.1.2.2 Validation of the DSA Primes p and q that were Constructed Using the ShaweTaylor Algorithm
The validation of the primes p and q that were generated by the method described in Appendix
A.1.2.1.2 may be performed if the values of firstseed, pseed, qseed, pgen_counter and
qgen_counter were saved and are provided for use in the following algorithm.
The following process or its equivalent shall be used to validate p and q for this constructive
method.
Input:
1. p, qThe primes to be validated.

2. firstseed, pseed, qseed

Seed values that were used to generate p and q.

3. pgen_counter, qgen_counter
The count values that were determined during generation.
Output:
1. status

The status returned from the validation procedure, where
status is either SUCCESS or FAILURE.

Process:
1. L = len (p).
2. N = len (q).
3. Check that the (L, N) pair is in the list of acceptable (L, N) pairs (see Section 4.2). If
the pair is not in the list, then return FAILURE.
4. If (firstseed < 2N–1), then return FAILURE.
5. If (2N ≤ q), then return FAILURE).
6. If (2L ≤ p), then return FAILURE.
7. If ((p – 1) mod q ≠ 0), then return FAILURE.
8. Using L, N and firstseed, perform the constructive prime generation procedure in
Appendix A.1.2.1.2 to obtain p_val, q_val, pseed_val, qseed_val, pgen_counter_val,
and qgen_counter_val. If FAILURE is returned, or if (q_val ≠ q) or (qseed_val ≠40

qseed) or (qgen_counter_val ≠ qgen_counter) or (p_val ≠ p) or (pseed_val ≠ pseed)or (pgen_counter_val ≠ pgen_counter), then return FAILURE.
9. Return SUCCESS.
A.2

Generation of the Generator g

The generator g depends on the values of p and q. Two methods for determining the generator g
are provided; one of these methods shall be used. The first method, discussed in Appendix
A.2.1, may be used when complete validation of the generator g is not required; it isrecommended that this method be used only when the party generating g is trusted to notdeliberately generate a g that has a potentially exploitable relationship to another generator g′.
For example, it must be hard to determine an exponential relationship between the generatorssuch that g = (g′)x mod p for a known value of x. (Note: Read (g′)x as g prime to the x.)
Appendix A.2.2 provides a method for partial validation when the method of generation in
Appendix A.2.1 is used. The second method for generating g, discussed in Appendix A.2.3, shallbe used when validation of the generator g is required; the method for the validation of agenerator determined using the method in Appendix A.2.3 is provided in Appendix A.2.4.
A.2.1 Unverifiable Generation of the Generator gThis method is used to determine a value for g, based on the values of p and q. It may be used
when validation of the generator g is not required. The correct generation of g cannot becompletely validated (see Appendix A.2.2). Note that this generation method for g was also
specified in previous versions of this Standard.
The following process or its equivalent shall be used to generate the generator g for this method.
Input:
1. p, qThe generated primes.

Output:
1. g

The requested value of g.

Process:
1. e = (p – 1)/q.
2. Set h = any integer satisfying 1 < h < ( p – 1), such that h differs from any value
previously tried. Note that h could be obtained from a random number generator orfrom a counter that changes after each use.
3. g = he mod p.

41

4. If (g = 1), then go to step 2.
5. Return g.
A.2.2 Assurance of the Validity of the Generator g
The order of the generator g that was generated using Appendix A.2.1 can be partially validated
by checking the range and order, thereby performing a partial validation of g.
The following process or its equivalent shall be used when partial validation of the generator g isrequired:
Input:
1. p, q, g

The domain parameters.

Output:
1. status

The status returned from the generation routine, where status is either
PARTIALLY VALID or INVALID.

Process:
1. Verify that 2 ≤ g ≤ (p–1). If not true, return INVALID.
2. If (gq = 1 mod p), then return PARTIALLY VALID.
3. Return INVALID.
The non-existence of a potentially exploitable relationship of g to another generator g′ (that is
known to the entity that generated g, but may not be known by other entities) cannot be checked.
In this sense, the correct generation of g cannot be completely validated.
A.2.3 Verifiable Canonical Generation of the Generator g
The generation of g is based on the values of p, q and domain_parameter_seed (which are
outputs of the generation processes in Appendix A.1). When p and q were generated using themethod in Appendix A.1.1.2, the domain_parameter_seed value must have been returned fromthe generation routine. When p and q were generated using the method in Appendix A.1.2.1, thefirstseed, pseed, and qseed values must have been returned from the generation routine; in thiscase, domain_parameter_seed = firstseed || pseed || qseed shall be used in the following process.
This method of generating a generator g can be validated (see Appendix A.2.4).
This generation method supports the generation of multiple values of g for specific values of pand q. The use of different values of g for the same p and q may be used to support key
separation; for example, using the g that is generated with index = 1 for digital signatures andwith index = 2 for key establishment.

42

Let Hash( ) be the hash function used to generate p and q (see Appendix A.1). The following
process or its equivalent shall be used to generate the generator g.
Input:
1. p, qThe primes.

2. domain_parameter_seed

The seed used during the generation of p and q.

3. index

The index to be used for generating g. index is a bit stringof length 8 that represents an unsigned integer.

Output:
1. status

The status returned from the generation routine, where status is either
VALID or INVALID.

2. g

The value of g that was generated.

Process:

Note: count is an unsigned 16-bit integer.
Comment: Check that a valid value of the index has
been provided (see above).

1. If (index is incorrect), then return INVALID.
2. N = len(q).
3. e = (p – 1)/q.
4. count = 0.
5. count = count + 1.
Comment: Check that count does not wrap aroundto 0.
6. If (count = 0), then return INVALID.
Comment: the length of thedomain_parameter_seed has already been checked.
“ggen” is the bit string 0x6767656E.
7. U = domain_parameter_seed || “ggen” || index || count.
8. W = Hash(U).
9. g = We mod p.
10. If (g < 2), then go to step 5.

Comment: If a generator has not been found.

11. Return VALID and the value of g.

43

A.2.4 Validation Routine when the Canonical Generation of the Generator gRoutine Was Used
This algorithm shall be used to validate the value of g that was generated using the process inAppendix A.2.3, based on the values of p, q, domain_parameter_seed, and the appropriate valueof index. It is assumed that the values of p and q have been previously validated according toAppendix A.1. Note that the method specified in Appendix A.2.3 for the generation of g was notincluded in previous versions of this Standard; therefore, this validation method is not
appropriate for that case.
The domain_parameter_seed is an output from the generation of p and q. When p and q weregenerated using the method in Appendix A.1.1.2, the domain_parameter_seed must have beenreturned from the generation routine and made available to the validating party. When p and q
were generated using the method in Appendix A.1.2.1, the firstseed, pseed, and qseed values
must have been returned from the generation routine and made available; firstseed, pseed, andqseed shall be concatenated to form the domain_parameter_seed used in the following process.
Let Hash( ) be the hash function used to generate g (i.e., the hash function also used to generatep and q).
The input index is the index number for the generator g. See Appendix A.2.3 for more details.
The following process or its equivalent shall be used to validate the generator g for this method.
Input:
1. p, qThe primes.

2. domain_parameter_seed

The seed used to generate p and q.

3. index

The index used in Appendix A.2.3 to generate x. index isa bit string of length 8 that represents an unsigned
integer.

4. g

The value of g to be validated.

Output:
1. status
Process:

The status returned from the generation routine, where
status is either VALID or INVALID.
Note: count is an unsigned 16-bit integer.
Comment: Check that a valid value of the index has been
provided (see above).

1. If (index is incorrect), then return INVALID.
2. Verify that 2 ≤ g ≤ (p–1). If not true, return INVALID.
3. If (gq ≠ 1 mod p), then return INVALID.

44

4. N = len(q).
5. e = (p – 1)/q.
6. count = 0.
7. count = count + 1.
Comment: Check that count does not wrap aroundto 0.
8. If (count = 0), then return INVALID.
Comment: “ggen” is the bit string 0x6767656E.
9. U = domain_parameter_seed || “ggen” || index || count.
10. W = Hash(U).
11. computed_g = We mod p.
12. If (computed_g < 2), then go to step 7. Comment: If a generator has not been found.
13. If (computed_g = g), then return VALID, else return INVALID.

45

APPENDIX B: Key Pair Generation
Discrete logarithm cryptography (DLC) is divided into finite field cryptography (FFC) andelliptic curve cryptography (ECC); the difference between the two is the type of math that isused. DSA is an example of FFC; ECDSA is an example of ECC. Other examples of DLC are
the Diffie-Hellman and MQV key agreement algorithms, which have both FFC and ECC forms.
The most common example of integer factorization cryptography (IFC) is RSA.
This appendix specifies methods for the generation of FFC and ECC key pairs and secret
numbers, and the generation of IFC key pairs. All generation methods require the use of an
approved, properly instantiated random bit generator (RBG); the RBG shall have a security
strength equal to or greater than the security strength associated with the key pairs and secret
numbers to be generated. See SP 800-57 for guidance on security strengths and key sizes.
This appendix does not indicate the required conversions between bit strings and integers. Whenrequired by a process in this appendix, the conversion shall be accomplished as specified inAppendix C.2.
B.1

FFC Key Pair Generation

An FFC key pair (x, y) is generated for a set of domain parameters (p, q, g {,domain_parameter_seed, counter}). Two methods are provided for the generation of the FFC
private key x and public key y; one of these two methods shall be used. Prior to generating DSA
key pairs, assurance of the validity of the domain parameters (p, q and g) shall have beenobtained as specified in Section 3.1.
For DSA, the valid values of L and N are provided in Section 4.2.
B.1.1 Key Pair Generation Using Extra Random Bits
In this method, 64 more bits are requested from the RBG than are needed for x so that bias
produced by the mod function in step 6 is negligible.
The following process or its equivalent may be used to generate an FFC key pair.
Input:(p, q, g)

The subset of the domain parameters that are used for this process. p, q
and g shall either be provided as integers during input, or shall be
converted to integers prior to use.

46Output:
1. status

The status returned from the key pair generation process. The status will
indicate SUCCESS or an ERROR.

2.

The generated private and public keys. If an error is encountered during
the generation process, invalid values for x and y should be returned, as
represented by Invalid_x and Invalid_y in the following specification. x
and y are returned as integers. The generated private key x is in the range[1, q–1], and the public key is in the range [1, p–1].

(x, y)

Process:
1. N = len(q); L = len(p).
Comment: Check that the (L, N) pair is specified in
Section 4.2.
2. If the (L, N) pair is invalid, then return an ERROR indicator, Invalid_x, andInvalid_y.
3. requested_security_strength = the security strength associated with the (L, N) pair;see SP 800-57.
4. Obtain a string of N+64 returned_bits from an RBG with a security strength ofrequested_security_strength or more. If an ERROR indication is returned, thenreturn an ERROR indication, Invalid_x, and Invalid_y.
5. Convert returned_bits to the (non-negative) integer c (see Appendix C.2.1).
6. x = (c mod (q–1)) + 1.

Comment: 0 ≤ c mod (q–1) ≤ q–2 and implies that1 ≤ x ≤ q–1.

7. y = gx mod p.
8. Return SUCCESS, x, and y.
B.1.2 Key Pair Generation by Testing Candidates
In this method, a random number is obtained and tested to determine that it will produce a valueof x in the correct range. If x is out-of-range, another random number is obtained (i.e., theprocess is iterated until an acceptable value of x is obtained.
The following process or its equivalent may be used to generate an FFC key pair.
Input:(p, q, g)

The subset of the domain parameters that are used for this process. p, q
and g shall either be provided as integers during input, or shall be
converted to integers prior to use.
47

Output:
1. status

The status returned from the key pair generation process. The status will
indicate SUCCESS or an ERROR.

2.

The generated private and public keys. If an error is encountered during
the generation process, invalid values for x and y should be returned, as
represented by Invalid_x and Invalid_y in the following specification. x
and y are returned as integers. The generated private key x is in the range[1, q–1], and the public key is in the range [1, p–1].

(x, y)

Process:
1. N = len(q); L = len(p).
Comment: Check that the (L, N) pair is specified in
Section 4.2.
2. If the (L, N) pair is invalid, then return an ERROR indication, Invalid_x, andInvalid_y.
3. requested_security_strength = the security strength associated with the (L, N) pair;see SP 800-57.
4. Obtain a string of N returned_bits from an RBG with a security strength ofrequested_security_strength or more. If an ERROR indication is returned, thenreturn an ERROR indication, Invalid_x, and Invalid_y.
5. Convert returned_bits to the (non-negative) integer c (see Appendix C.2.1).
6. If (c > q–2), then go to step 4.
7. x = c + 1.
8. y = gx mod p.
9. Return SUCCESS, x, and y.
B.2

FFC Per-Message Secret Number Generation

DSA requires the generation of a new random number k for each message to be signed. Two
methods are provided for the generation of k; one of these two methods or another approved
method shall be used.
The valid values of N are provided in Section 4.2. Let inverse(k, q) be a function that computes
the inverse of a (non-negative) integer k with respect to multiplication modulo the prime number
q. A technique for computing the inverse is provided in Appendix C.1.

48

B.2.1 Per-Message Secret Number Generation Using Extra Random Bits
In this method, 64 more bits are requested from the RBG than are needed for k so that bias
produced by the mod function in step 6 is not readily apparent.
The following process or its equivalent may be used to generate a per-message secret number.
Input:(p, q, g)

DSA domain parameters that are generated as specified in Section 4.3.1.

Output:
1. status
2.

(k, k −1 )The status returned from the secret number generation process. The statuswill indicate SUCCESS or an ERROR.
The per-message secret number k and its mod q inverse, k −1 . If an error isencountered during the generation process, invalid values for k and k −1
should be returned, as represented by Invalid_k and Invalid_k_inverse in
the following specification. k and k −1 are in the range [1, q–1].

Process:
1. N = len(q); L = len(p).
Comment: Check that the (L, N) pair is specified in
Section 4.2.
2. If the (L, N) pair is invalid, then return an ERROR indication, Invalid_k, andInvalid_k_inverse.
3. requested_security_strength = the security strength associated with the (L, N) pair;see SP 800-57.
4. Obtain a string of N+64 returned_bits from an RBG with a security strength ofrequested_security_strength or more. If an ERROR indication is returned, thenreturn an ERROR indication, Invalid_k, and Invalid_k_inverse.
5. Convert returned_bits to the (non-negative) integer c (see Appendix C.2.1).
6. k = (c mod (q–1)) + 1.
7. (status, k–1) = inverse (k, q).
8. Return status, k, and k −1 .
B.2.2 Per-Message Secret Number Generation by Testing Candidates
In this method, a random number is obtained and tested to determine that it will produce a valueof k in the correct range. If k is out-of-range, another random number is obtained (i.e., theprocess is iterated until an acceptable value of k is obtained.
49

The following process or its equivalent may be used to generate a per-message secret number.
Input:(p, q, g)

DSA domain parameters that are generated as specified in Section 4.3.1.

Output:
1. status
2.

(k, k −1 )The status returned from the secret number generation process. The statuswill indicate SUCCESS or an ERROR.
The per-message secret number k and its inverse, k −1 . If an error isencountered during the generation process, invalid values for k andk −1 should be returned, as represented by Invalid_k and Invalid_k_inversein the following specification. k and k −1 are in the range [1, q–1].

Process:
1. N = len(q); L = len(p).
Comment: Check that the (L, N) pair is specified in
Section 4.2).
2. If the (L, N) pair is invalid, then return an ERROR indication, Invalid_k, andInvalid_k_inverse.
3. requested_security_strength = the security strength associated with the (L, N) pair;see SP 800-57.
4. Obtain a string of N returned_bits from an RBG with a security strength ofrequested_security_strength or more. If an ERROR indication is returned, thenreturn an ERROR indication, Invalid_k, and Invalid_k_inverse.
5. Convert returned_bits to the (non-negative) integer c (see Appendix C.2.1).
6. If (c > q–2), then go to step 4.
7. k = c + 1.
8. (status, k −1 ) = inverse(k, q).
9. Return status, k, and k −1 .
B.3

IFC Key Pair Generation

B.3.1 Criteria for IFC Key Pairs
Key pairs for IFC consist of a public key (n, e), and a private key (n, d), where n is the modulus
and is the product of two prime numbers p and q. The security of IFC depends on the quality andsecrecy of these primes and the private exponent d. The primes p and q shall be generated using
50

one of the following methods:
A. Both p and q are randomly generated prime numbers (Random Primes), where p and q
shall both be either :1. Provable primes (see Appendix B.3.2), or2.

Probable primes (see Appendix B.3.3).

Using methods 1 and 2, p and q with lengths of 1024 or 1536 bits may be generated; pand q with lengths of 512 bits shall not be generated using these methods. Instead, p
and q with lengths of 512 bits shall be generated using the conditions based on auxiliary
primes (see Appendices B.3.4, B.3.5, or B.3.6).
B. Both p and q are randomly generated prime numbers that satisfy the following additional
conditions (Primes with Conditions):•(p–1) has a prime factor p 1•(p+1) has a prime factor p 2•(q–1) has a prime factor q 1•(q+1) has a prime factor q 2

where p 1 , p 2 , q 1 and q 2 are called auxiliary primes of p and q.
Using this method, one of the following cases shall apply:
1. The primes p 1 , p 2 , q 1 , q 2 , p and q shall all be provable primes (see Appendix
B.3.4),
2. The primes p 1 , p 2 , q 1 and q 2 shall be provable primes, and the primes p and q
shall be probable primes (see Appendix B.3.5), or3

The primes p 1 , p 2 , q 1 , q 2 , p and q shall all be probable primes (see Appendix
B.3.6).

The minimum lengths for each of the auxiliary primes p 1 , p 2 , q 1 and q 2 are dependent on
nlen, where nlen is the length of the modulus n in bits. Note that nlen is also called the
key size. The lengths of the auxiliary primes may be fixed or randomly chosen, subject tothe restrictions in Table B.1. The maximum length is determined by nlen (the sum of thelength of each auxiliary prime pair) and whether the primes p and q are probable primes
or provable primes (e.g., for the auxiliary prime pair p 1 and p 2 , len(p 1 ) + len(p 2 ) shall beless than a value determined by nlen, whether p 1 and p 2 are generated to be probable orprovable primes) 3.

3

For the probable primes p and q: len(p1) + len(p2) < len(p) – log2(len(p)) – 6; similarly for len(q1) + len(q2) andlen(q). For the provable primes p and q: len(p1) + len(p2) < len(p)/2 – log2(len(p)) – 7; similarly for len(q1) + len(q2)51Table B.1. Minimum and maximum lengths of p 1 , p 2 , q 1 and q 2
nlen

Min. length of
auxiliary primes
p 1 , p 2 , q 1 and q 2

Max. length of len(p 1 ) + len(p 2 ) andlen(q 1 ) + len(q 2 )p, q Probable primes

p, q Provable primes

1024> 100 bits

< 496 bits

< 239 bits

2048

> 140 bits

< 1007 bits

< 494 bits3072

> 170 bits

< 1518 bits

< 750 bits

For different values of nlen (i.e., different key sizes), the methods allowed for the generation of
p and q are specified in Table B.2.
Table B.2. Allowable Prime Generation Methods
nlen

Random Primes

Primes with Conditions

1024

No

Yes

2048

Yes

Yes

3072

Yes

Yes

In addition, all IFC keys shall meet the following criteria in order to conform to FIPS 186-4:
1. The public exponent e shall be selected with the following constraints:(a) The public verification exponent e shall be selected prior to generating the primes
p and q, and the private signature exponent d.
(b) The exponent e shall be an odd positive integer such that:
216 < e < 2256.
Note that the value of e may be any value that meets constraint 1(b), i.e., e may beeither a fixed value or a random value.
2. The primes p and q shall be selected with the following constraints:(a) (p–1) and (q–1) shall be relatively prime to the public exponent e.
(b) The private prime factor p shall be selected randomly and shall satisfy

and len(q). In each case, len(p) = len(q) = nlen/2.

52

( 2 )(2(nlen / 2) – 1) ≤ p ≤ (2nlen / 2– 1), where nlen is the appropriate length for thedesired security_strength.
(c) The private prime factor q shall be selected randomly and shall satisfy( 2 )(2(nlen / 2) – 1) ≤ q ≤ (2nlen / 2– 1), where nlen is the appropriate length for thedesired security_strength.
(d) |p – q| > 2(nlen / 2) – 100.
3. The private signature exponent d shall be selected with the following constraints after thegeneration of p and q:(a) The exponent d shall be a positive integer value such that
2nlen/ 2 < d < LCM(p–1, q–1), and(b) d = e–1 mod (LCM(p–1, q–1)).
That is, the inequality in (a) holds, and 1 ≡ (ed) (mod LCM(p–1, q–1)).
In the extremely rare event that d ≤ 2nlen / 2, then new values for p, q and d shall be
determined. A different value of e may be used, although this is not required.
Any hash function used during the generation of the key pair shall be approved (i.e., specified in
FIPS 180).
B.3.2 Generation of Random Primes that are Provably Prime
An approved method that satisfies the constraints of Appendix B.3.1 shall be used for thegeneration of IFC random primes p and q that are provably prime (see case A.1). One suchmethod is provided in Appendix B.3.2.1 and B.3.2.2. For this method, a random seed is initiallyrequired (see Appendix B.3.2.1); the length of the seed is equal to twice the security strength
associated with the modulus n. After the seed is obtained, the primes can be generated (seeAppendix B.3.2.2).
B.3.2.1 Get the Seed
The following process or its equivalent shall be used to generate the seed for this method.
Input:
nlen

The intended bit length of the modulus n.

Output:status

The status to be returned, where status is either SUCCESS or FAILURE.

seed

The seed. If status = FAILURE, a value of zero is returned as the seed.

Process:
1. If nlen is not valid (see Section 5.1), then Return (FAILURE, 0).
53

2. Let security_strength be the security strength associated with nlen, as specified in SP
800-57, Part 1.
3. Obtain a string seed of (2 * security_strength) bits from an RBG that supports the
security_strength.
4. Return (SUCCESS, seed).
B.3.2.2 Construction of the Provable Primes p and q
The following process or its equivalent shall be used to construct the random primes p and q (tobe used as factors of the RSA modulus n) that are provably prime:
Input:
nlen

The intended bit length of the modulus n.

e

The public verification exponent.

seed

The seed obtained using the method in Appendix B.3.2.1.

Output:status

The status of the generation process, where status is either SUCCESS or
FAILURE. When FAILURE is returned, zero values shall be returned as theother parameters.

p and qThe private prime factors of n.

Process:
1. If nlen is neither 2048 nor 3072, then return (FAILURE, 0, 0).
2. If ((e ≤ 216) OR (e ≥ 2256) OR (e is not odd)), then return (FAILURE, 0, 0).
3. Set the value of security_strength in accordance with the value of nlen, as specified in SP
800-57, Part 1.
4. If (len(seed) ≠ 2 * security_strength), then return (FAILURE, 0, 0).
5. working_seed = seed.
6. Generate p:
6.1

Using L = nlen/2, N 1 = 1, N 2 = 1, first_seed = working_seed and e, use the provableprime construction method in Appendix C.10 to obtain p and pseed. If FAILURE
is returned, then return (FAILURE, 0, 0).

6.2

working_seed = pseed.

54

7. Generate q:
7.1

Using L = nlen/2, N 1 = 1, N 2 = 1, first_seed = working_seed and e, use the provableprime construction method in Appendix C.10 to obtain q and qseed. If FAILURE
is returned, then return (FAILURE, 0, 0).

7.2

working_seed = qseed.

8. If ( |p – q| ≤ 2nlen/2 – 100), then go to step 7.
9. Zeroize the internally generated seeds:
9.1 pseed = 0;
9.2 qseed = 0;
9.3 working_seed = 0.
10. Return (SUCCESS, p, q).
B.3.3 Generation of Random Primes that are Probably Prime
An approved method that satisfies the constraints of Appendix B.3.1 shall be used for thegeneration of IFC random primes p and q that are probably prime (see case A.2).
The following process or its equivalent shall be used to construct the random probable primes p
and q (to be used as factors of the RSA modulus n):
Input:
nlen

The intended bit length of the modulus n.

e

The public verification exponent.

Output:status

The status of the generation process, where status is either SUCCESS or
FAILURE.

p and qThe private prime factors of n. When FAILURE is returned, zero values shall be
returned as p and q.

Process:
1. If nlen is neither 2048 nor 3072, return (FAILURE, 0, 0).
2. If ((e ≤ 216) OR (e ≥ 2256) OR (e is not odd)), then return (FAILURE, 0, 0).
3. Set the value of security_strength in accordance with the value of nlen, as specified in SP
800-57, Part 1.
4. Generate p:
4.1

i = 0.
55

4.2

Obtain a string p of (nlen/2) bits from an RBG that supports the security_strength.

4.3

If (p is not odd), then p = p + 1.

4.4

If ((p < ( 2 )(2(nlen / 2) – 1)), then go to step 4.2.

4.5

If (GCD(p−1, e) = 1), then
4.5.1 Test p for primality as specified in Appendix C.3, using an appropriate
value from Table C-2 or C-3 in Appendix C.3 as the number of iterations.
4.5.2 If p is PROBABLY PRIME, then go to step 5.

4.6

i= i + 1.

4.7

If (i ≥ 5(nlen/2)), then return (FAILURE, 0, 0)
Else go to step 4.2.

5. Generate q:
5.1

i = 0.

5.2

Obtain a string q of (nlen/2) bits from an RBG that supports the security_strength

5.3

If (q is not odd), then q = q + 1.

5.4

If (|p – q| ≤ 2nlen/2 – 100), then go to step 5.2.

5.5

If ((q < ( 2 )(2(nlen / 2) – 1)), then go to step 5.2.

5.6

If (GCD(q−1, e) = 1) then
5.6.1 Test q for primality as specified in Appendix C.3, using an appropriate
value from Table C-2 or C-3 in Appendix C.3 as the number of iterations.
5.6.2 If q is PROBABLY PRIME, then return (SUCCESS, p, q).

5.7

i = i + 1.

5.8

If (i ≥ 5(nlen/2)), then return (FAILURE, 0, 0)
Else go to step 5.2.

B.3.4 Generation of Provable Primes with Conditions Based on Auxiliary
Provable Primes
This section specifies an approved method for the generation of the IFC primes p and q with theadditional conditions specified in Appendix B.3.1, case B.1, where p, p 1 , p 2 , q, q 1 and q 2 are allprovable primes. For this method, a random seed is initially required (see Appendix B.3.2.1); thelength of the seed is equal to twice the security strength associated with the modulus n. After the
first seed is obtained, the primes can be generated.

56

Let bitlen 1 , bitlen 2 , bitlen 3 , and bitlen 4 be the bit lengths for p 1 , p 2 , q 1 and q 2 , respectively, in
accordance with Table B.1. The following process or its equivalent shall be used to generate theprovable primes:
Input:
nlen

The intended bit length of the modulus n.

e

The public verification exponent.

seed

The seed obtained using the method in Appendix B.3.2.1.

Output:status

The status of the generation process, where status is either SUCCESS or
FAILURE. If FAILURE is returned then zeros shall be returned as the values
for p and q.

p and qThe private prime factors of n.

Process:
1. If nlen is neither 1024, 2048, nor 3072, then return (FAILURE, 0, 0).
2. If ((e ≤ 216) OR (e ≥ 2256) OR (e is not odd)), then return (FAILURE, 0, 0).
3. Set the value of security_strength in accordance with the value of nlen, as specified in SP
800-57, Part 1.
4. If (len(seed) ≠ 2 * security_strength), then return (FAILURE, 0, 0).
5. working_seed = seed.
6. Generate p:
6.1 Using L = nlen/2, N 1 = bitlen 1 , N 2 = bitlen 2 , firstseed = working_seed and e, use theprovable prime construction method in Appendix C.10 to obtain p, p 1 , p 2 and pseed.
If FAILURE is returned, return (FAILURE, 0, 0).
6.2 working_seed = pseed.
7. Generate q:
7.1 Using L = nlen/2, N 1 = bitlen 3 , N 2 = bitlen 4 and firstseed = working_seed and e, use
the provable prime construction method in Appendix C.10 to obtain q, q 1 , q 2 andqseed. If FAILURE is returned, return (FAILURE, 0, 0).
7.2 working_seed = qseed.
8. If ( |p – q| ≤ 2nlen/2 – 100), then go to step 7.
9. Zeroize the internally generated seeds:
9.1 pseed = 0.
57

9.2 qseed = 0.
9.3 working_seed = 0.
10. Return (SUCCESS, p, q).

B.3.5 Generation of Probable Primes with Conditions Based on Auxiliary
Provable Primes
This section specifies an approved method for the generation of the IFC primes p and q with theadditional conditions specified in Appendix B.3.1, case B.2, where p 1 , p 2 , q 1 and q 2 are provably
prime, and p and q are probably prime. For this method, a random seed is initially required (seeAppendix B.3.2.1); the length of the seed is equal to twice the security strength associated withthe modulus n. After the first seed is obtained, the primes can be generated.
Let bitlen 1 , bitlen 2 , bitlen 3 , and bitlen 4 be the bit lengths for p 1 , p 2 , q 1 and q 2 , respectively inaccordance with Table B.1. The following process or its equivalent shall be used to construct p
and q.
Input:
nlen

The intended bit length of the modulus n.

e

The public verification exponent.

seed

The seed obtained using the method in Appendix B.3.2.1.

Output:status

The status of the generation process, where status is either SUCCESS or
FAILURE. If FAILURE is returned then zeros shall be returned as thevalues for p and q.

p and qThe private prime factors of n.

Process:
1. If nlen is neither 1024, 2048, nor 3072, then return (FAILURE, 0, 0).
2. If ((e ≤ 216) OR (e ≥ 2256) OR (e is not odd)), then return (FAILURE, 0, 0).
3. Set the value of security_strength in accordance with the value of nlen, as specified in
SP 800-57, Part 1.
4. If (len(seed) ≠ 2 * security_strength), then return (FAILURE, 0, 0).
Comment: Generate four primes p 1 , p 2 , q 1 and q 2that are provably prime.

58

5. Generate p:
5.1

Using bitlen 1 as the length, and seed as the input_seed, use the random prime
generation routine in Appendix C.6 to obtain p 1 and prime_seed. If FAILURE
is returned, the return (FAILURE, 0, 0).

5.2 Using bitlen 2 as the length, and prime_seed as the input_seed, use the random
prime generation routine in Appendix C.6 to obtain p 2 and a new value forprime_seed. If FAILURE is returned, the return (FAILURE, 0, 0).
5.3

Generate a prime p using the routine in Appendix C.9 with inputs of p 1 , p 2 ,nlen, e and security_strength, also obtaining X p . If FAILURE is returned,
return (FAILURE, 0, 0).

6. Generate q:
6.1. Using bitlen 3 as the length, and prime_seed as the input_seed, use the random
prime generation routine in Appendix C.6 to obtain q 1 and a new value forprime_seed. If FAILURE is returned, the return (FAILURE, 0, 0).
6.2

Using bitlen 4 as the length, and prime_seed as the input_seed, use the random
prime generation routine in Appendix C.6 to obtain q 2 and a new value forprime_seed. If FAILURE is returned, the return (FAILURE, 0, 0).

6.3

Generate a prime q using the routine in Appendix C.9 with inputs of q 1 , q 2 ,nlen, e and security_strength, also obtaining X q . If FAILURE is returned,
return (FAILURE, 0, 0).

7. If ((|p – q| ≤ 2nlen/2 –100) OR (|X p – X q | ≤ 2nlen/2 – 100)), then go to step 6.
8. Zeroize the internally generated that are not returned:
8.1

X p = 0.

8.2

X q = 0.

8.3

prime_seed = 0.

8.4

p 1 = 0.

8.5

p 2 = 0.

8.6

q 1 = 0.

8.7

q 2 = 0.

9. Return (SUCCESS, p, q).

59

B.3.6 Generation of Probable Primes with Conditions Based on Auxiliary
Probable Primes
An approved method that satisfies the constraints of Appendix B.3.1 shall be used for thegeneration of IFC primes p and q that are probably prime and meet the additional constraints ofAppendix B.3.1 (see case B.3). For this case, the prime factors p 1 , p 2 , q 1 and q 2 are also
probably prime.
Four random numbers X p1 , X p2 , X q1 and X q2 are generated, from which the prime factors p 1 , p 2 ,q 1 and q 2 are determined. p 1 and p 2 , and an additional random number X p are then used todetermine p, and q 1 and q 2 and a random number X q are used to obtain q. Let bitlen 1 , bitlen 2 ,bitlen 3 , and bitlen 4 be the bit lengths for p 1 , p 2 , q 1 and q 2 , respectively chosen in accordancewith Table B.1.
The following process or its equivalent shall be used to generate p and q:
Input:
nlen

The intended bit length of the modulus n.

e

The public verification exponent.

Output:status

The status of the generation process, where status is either SUCCESS or
FAILURE. If FAILURE is returned then zeros shall be returned as thevalues for p and q.

p and qThe private prime factors of n.

Process:
1. If nlen is neither 1024, 2048, nor 3072, then return (FAILURE, 0, 0).
2. If ((e ≤ 216) OR (e ≥ 2256) OR (e is not odd)), then return (FAILURE, 0, 0).
3. Set the value of security_strength in accordance with the value of nlen, as specified in
SP 800-57, Part 1.
4. Generate p:
4.1

Generate an odd integer X p1 of length bitlen 1 bits, and a second odd integer X p2of length bitlen 2 bits, using an approved random number generator that supports
the security_strength.

4.2

Sequentially search successive odd integers, starting at X p1 until the firstprobable prime p 1 is found. Candidate integers shall be tested for primality as
specified in Appendix C.3. Repeat the process to find p 2 , starting at X p2 . The
probable primes p 1 and p 2 shall be the first integers that pass the primality test.

60

4.3

Generate a prime p using the routine in Appendix C.9 with inputs of p 1 , p 2 ,nlen, e and security_ strength, also obtaining X p . If FAILURE is returned,
return (FAILURE, 0, 0).

5. Generate q:
5.1

Generate an odd integer X q1 of length bitlen 3 bits, and a second odd integer X q2of length bitlen 4 bits, using an approved random number generator that supports
the security_strength.

5.2

Sequentially search successive odd integers, starting at X q1 until the firstprobable prime q 1 is found. Candidate integers shall be tested for primality as
specified in Appendix C.3. Repeat the process to find q 2 , starting at X q2 . The
probable primes q 1 and q 2 shall be the first integers that pass the primality test.

5.3

Generate a prime q using the routine in Appendix C.9 with inputs of q 1 , q 2 ,nlen, e and security_ strength, also obtaining X q . If FAILURE is returned,
return (FAILURE, 0, 0).

6. If ((|X p – X q | ≤ 2nlen/2 –100) OR (|p – q| ≤ 2nlen/2 – 100))), then go to step 5.
7. Zeroize the internally generated values that are not returned:
7.1

X p = 0.

7.2

X q = 0.

7.3

X p1 = 0.

7.4

X p2 = 0.

7.5

X q1 = 0.

7.6

X q2 = 0.

7.7

p 1 = 0.

7.8

p 2 = 0.

7.9

q 1 = 0.

7.10 q 2 = 0.
8. Return (SUCCESS, p, q).
B.4

ECC Key Pair Generation

An ECC key pair d and Q is generated for a set of domain parameters (q, FR, a, b {,domain_parameter_seed}, G, n, h). Two methods are provided for the generation of the ECC
private key d and public key Q; one of these two methods shall be used to generate d and Q.

61

Prior to generating ECDSA key pairs, assurance of the validity of the domain parameters (q, FR,a, b {, domain_parameter_seed}, G, n, h) shall have been obtained as specified in Section 3.1.
For ECDSA, the valid bit-lengths of n are provided in Section 6.1.1. See ANS X9.62 fordefinitions of the elliptic curve math and the conversion routines.
B.4.1 Key Pair Generation Using Extra Random Bits
In this method, 64 more bits are requested from the RBG than are needed for d so that bias
produced by the mod function in step 6 is negligible.
The following process or its equivalent may be used to generate an ECC key pair.
Input:
1. (q, FR, a, b {, domain_parameter_seed}, G, n, h)
The domain parameters that are used for this process. n is a prime number,and G is a point on the elliptic curve.
Output:
1. status

The status returned from the key pair generation procedure. The status will
indicate SUCCESS or an ERROR.

2.

The generated private and public keys. If an error is encountered during
the generation process, invalid values for d and Q should be returned, as
represented by Invalid_d and Invalid_Q in the following specification. d isan integer, and Q is an elliptic curve point. The generated private key d isin the range [1, n–1].

(d, Q)

Process:
1. N = len(n).
Comment: Check that N is included in Table 1 ofSection 6.1.1.
2. If N is invalid, then return an ERROR indication, Invalid_d, and Invalid_Q.
3. requested_security_strength = the security strength associated with N; see SP 800-57,
Part 1.
4. Obtain a string of N+64 returned_bits from an RBG with a security strength ofrequested_security_strength or more. If an ERROR indication is returned, thenreturn an ERROR indication, Invalid_d, and Invalid_Q.
5. Convert returned_bits to the (non-negative) integer c (see Appendix C.2.1).
6. d = (c mod (n–1)) + 1.

62

7. Q = dG.
8. Return SUCCESS, d, and Q.
B.4.2 Key Pair Generation by Testing Candidates
In this method, a random number is obtained and tested to determine that it will produce a valueof d in the correct range. If d is out-of-range, another random number is obtained (i.e., theprocess is iterated until an acceptable value of d is obtained.
The following process or its equivalent may be used to generate an ECC key pair.
Input:
1. (q, FR, a, b {, domain_parameter_seed}, G, n, h)
The domain parameters that are used for this process. n is a prime number,and G is a point on the elliptic curve.
Output:
1. status

The status returned from the key pair generation procedure. The status will
indicate SUCCESS or an ERROR.

2.

The generated private and public keys. If an error is encountered during
the generation process, invalid values for d and Q should be returned, as
represented by Invalid_d and Invalid_Q in the following specification. d isan integer, and Q is an elliptic curve point. The generated private key d isin the range [1, n–1].

(d, Q)

Process:
1. N = len(n).
Comment: Check that N is included in Table 1 ofSection 6.1.1.
2. If N is invalid, then return an ERROR indication, Invalid_d, and Invalid_Q.
3. requested_security_strength = the security strength associated with N; see SP 800-57,
Part 1.
4. Obtain a string of N returned_bits from an RBG with a security strength ofrequested_security_strength or more. If an ERROR indication is returned, thenreturn an ERROR indication, Invalid_d, and Invalid_Q.
5. Convert returned_bits to the (non-negative) integer c (see Appendix C.2.1).
6. If (c > n–2), then go to step 4.
7. d = c + 1.
63

8. Q = dG.
9. Return SUCCESS, d, and Q.
B.5

ECC Per-Message Secret Number Generation

ECDSA requires the generation of a new random number k for each message to be signed. Two
methods are provided for the generation of k; one of these two methods or another approved
method shall be used.
The valid values of n are provided in Section 6.1.1. See ANS X9.62 for definitions of the elliptic
curve math and the conversion routines.
Let inverse(k, n) be a function that computes the inverse of a (non-negative) integer k withrespect to multiplication modulo the prime number n. A technique for computing the inverse isprovided in Appendix C.1.
B.5.1 Per-Message Secret Number Generation Using Extra Random Bits
In this method, 64 more bits are requested from the RBG than are needed for k so that bias
produced by the mod function in step 6 is not readily apparent.
The following process or its equivalent may be used to generate a per-message secret number.
Input:
1.

(q, FR, a, b {, domain_parameter_seed}, G, n, h)
The domain parameters that are used for this process. n is a prime
number, and G is a point on the elliptic curve.

Output:
1. status
2.

(k, k −1 )The status returned from the key pair generation procedure. The status will
indicate SUCCESS or an ERROR.
The generated secret number k and its inverse k–1. If an error isencountered during the generation process, invalid values for k andk −1 should be returned, as represented by Invalid_k and Invalid_k_inversein the following specification. k and k −1 are integers in the range [1, n–1].

Process:
1. N = len(n).
Comment: Check that N is included in Table 1 ofSection 6.1.1.
2. If N is invalid, then return an ERROR indication, Invalid_k, and Invalid_k_inverse.

643. requested_security_strength = the security strength associated with N; see SP 800-57,
Part 1.
4. Obtain a string of N+64 returned_bits from an RBG with a security strength ofrequested_security_strength or more. If an ERROR indication is returned, thenreturn an ERROR indication, Invalid_k, and Invalid_k_inverse.
5. Convert returned_bits to the non-negative integer c (see Appendix C.2.1).
6. k = (c mod (n–1)) + 1.
7. (status, k −1 ) = inverse(k, n).
8. Return status, k, and k −1 .
B.5.2 Per-Message Secret Number Generation by Testing Candidates
In this method, a random number is obtained and tested to determine that it will produce a valueof k in the correct range. If k is out-of-range, another random number is obtained (i.e., theprocess is iterated until an acceptable value of k is obtained.
The following process or its equivalent may b used to generate a per-message secret number.
Input:
1.

(q, FR, a, b {, domain_parameter_seed}, G, n, h)
The domain parameters that are used for this process. n is a prime number,and G is a point on the elliptic curve.

Output:
1. status
2.

(k, k −1 )The status returned from the key pair generation procedure. The status will
indicate SUCCESS or an ERROR.
The generated secret number k and its inverse k–1. If an error isencountered during the generation process, invalid values for k and k −1
should be returned, as represented by Invalid_k and Invalid_k_inverse in
the following specification. k and k −1 are integers in the range [1, n–1].

Process:
1. N = len(n).
Comment: Check that N is included in Table 1 ofSection 6.1.1.
2. If N is not included in Table 1, then return an ERROR indication, Invalid_k, andInvalid_k_inverse.

65

3. requested_security_strength = the security strength associated with N; see SP 800-57,
Part 1.
4. Obtain a string of N returned_bits from an RBG with a security strength ofrequested_security_strength or more. If an ERROR indication is returned, thenreturn an ERROR indication, Invalid_k, and Invalid_k_inverse.
5. Convert returned_bits to the (non-negative) integer c (see Appendix C.2.1).
6. If (c > n–2), then go to step 4.
7. k = c + 1.
8. (status, k–1) = inverse(k, n).
9. Return status, k, and k −1 .

66Appendix C: Generation of Other Quantities
This appendix contains routines for supplementary processes required for the implementation ofthis Standard. Appendix C.1 is needed to produce the inverse of the per-message secret k (seeSection 4.5, and Appendices B.2.1, B.2.2, B.5.1 and B.5.2) and the inverse of the signature
portion s that is used during signature verification (see Section 4.7). The routines in Appendix
C.2 are required to convert between bit strings and integers where required in implementing thisStandard. Appendix C.3 contains probabilistic primality tests to be used during the generation of
DSA domain parameters and RSA key pairs. Appendices C.4 and C.5 contain algorithms
required during the Lucas probabilistic primality test of Appendix C.3.3 to check for a perfectsquare and to compute the Jacobi symbol. Appendix C.6 contains the Shawe-Taylor algorithm
for the construction of primes. Appendix C.7 provides a process to perform trial division, as
required by the random prime generation routine in Appendix C.6. The sieve procedure in
Appendix C.8 is needed by the trial division routine in Appendix C.7. The trial division process
in Appendix C.7 and the sieve procedure in Appendix C.8 have been extracted from ANS X9.80,
Prime Number Generation, Primality Testing, and Primality Certificates. Appendix C.9 isrequired during the generation of RSA key pairs. Appendix C.10 provides a method forconstructing provable primes for RSA (see Appendix B.3.2.2 and B.3.4).
C.1

Computation of the Inverse Value

This algorithm or an algorithm that produces an equivalent result shall be used to compute themultiplicative inverse z–1 mod a, where 0 < z < a, 0 < z–1 < a, and a is a prime number. In thisStandard, z is either k or s, and a is either q or n.
Input:
1. z

The value to be inverted mod a (i.e., either k or s).

2. a

The domain parameter and (prime) modulus (i.e., either q or n).

Output:
1. status

The status returned from this function, where the status is either
SUCCESS or ERROR.

2. z–1

The multiplicative inverse of z mod a, if it exists.

Process:
1. Verify that a and z are positive integers such that z < a; if not, return an ERROR
indication.
2. Set i = a, j = z, y 2 = 0, and y 1 = 1.
3. quotient =  i/j.
67

4. remainder = i –( j * quotient).
5. y = y 2 –(y 1 * quotient).
6. Set i = j, j = remainder, y 2 = y 1 , and y 1 = y.
7. If (j > 0), then go to step 3.
8. If (i ≠ 1), then return an ERROR indication.
9. Return SUCCESS and y 2 mod a.
C.2

Conversion Between Bit Strings and Integers

C.2.1 Conversion of a Bit String to an Integer
An n-long sequence of bits { x 1 , …, xn } is converted to an integer by the rule
{ x 1 , … , x n } → (x 1 ∗ 2

n–1

n–2

) + (x 2 ∗ 2) + … + (n 1 ∗ 2) + x n .

Note that the first bit of a sequence corresponds to the most significant bit of the corresponding
integer, and the last bit corresponds to the least significant bit.
Input:
1. b 1 , b 2 , … , b n

The bit string to be converted.

Output:
1. C

The requested integer representation of the bit string.

Process:
1. Let (b 1 , b 2 , … , b n ) be the bits of b from leftmost to rightmost.
n

2. C = ∑ 2( n−i ) bi
i =1

3. Return C.
In this Standard, the binary length of an integer C is defined as the smallest integer n satisfying C< 2n .
C.2.2 Conversion of an Integer to a Bit String
An integer x in the range 0 ≤ x < 2n may be converted to an n-long sequence of bits by using its
binary expansion as shown below:68x = (x 1 ∗ 2n–1) + (x 2 ∗ 2n–2) + … + (x n–1 ∗ 2) + x n → {x 1 , … , x n }Note that the first bit of a sequence corresponds to the most significant bit of the corresponding
integer, and the last bit corresponds to the least significant bit.
Input:
1. C

The non-negative integer to be converted.

Output:
1. b 1 , b 2 , …, b n The bit string representation of the integer C.
Process:
1. Let (b 1 , b 2 , …, b n ) represent the bit string, where b i = 0 or 1, and b 1 is the mostsignificant bit, while b n is the least significant bit.
2. For any integer n that satisfies C < 2n, the bits b i shall satisfy:nC = ∑ 2 ( n −i ) bi
i =1

3. Return b 1 , b 2 , …, b n .
In this Standard, the binary length of the integer C is defined as the smallest integer n thatsatisfies C < 2n.
C.3

Probabilistic Primality Tests

A probabilistic primality test may be required during the generation and validation of prime
numbers. An approved robust probabilistic primality test shall be selected and used.
There are several probabilistic algorithms available. The Miller-Rabin probabilistic primality
tests described in Appendices C.3.1 and C.3.2 are versions of a procedure due to M.O. Rabin,based in part on ideas of Gary L. Miller; one of these versions shall be used as the Miller-Rabin
test discussed below. For more information, see [4]. For these tests, let RBG be an approved
random bit generator.
There are several Lucas probabilistic primality tests available; the version provided in [5] is
specified in Appendix C.3.3.
This Standard allows two alternatives for testing primality: either using several iterations of only
the Miller-Rabin test, or using the iterated Miller-Rabin test, followed by a single Lucas test. The
value of iterations (as used in Appendices C.3.1 and C.3.2) depends on the algorithm being used,
the security strength, the error probability used, the length (in bits) of the candidate prime and69

the type of tests to be performed. Tables C.1, C.2 and C.3 list the minimum number of iterationsof the Miller-Rabin tests that shall be performed.
As stated in Appendix F, if the definition of the error probability that led to the values of thenumber of Miller-Rabin tests for p and q in Tables C.1, C.2 and C.3 is not conservative enough,the prescribed number of Miller-Rabin tests can be followed by a single Lucas test. Since there
are no known non-prime values that pass the two test combination (i.e., the indicated number ofrounds of the Miller-Rabin test with randomly selected bases, followed by one round of theLucas test), the two test combination may provide additional assurance of primality over the useof only the Miller-Rabin test. For DSA, the two-test combination may provide better
performance. However, the Lucas test is not required when testing the p 1 , p 2 , q 1 and q 2 values
for primality when generating RSA primes. See Appendix F for further information.
Table C.1. Minimum number of Miller-Rabin iterations for DSA
Parameters
p: 1024 bits
q: 160 bits
Error probability = 2 −80
p: 2048 bits
q: 224 bits

M-R Tests Only

For p and q: 40

For p and q: 56

For p: 3
For q: 24

For p and q: 56

For p: 3
For q: 27

Error probability = 2 −112
p: 3072 bits
q: 256 bits

For p: 3
For q: 19

Error probability = 2 −112
p: 2048 bits
q: 256 bits

M-R Tests when followedby One Lucas test

For p and q: 64

For p: 2
For q: 27

Error probability = 2 −128

70

Table C.2. Minimum number of rounds of M-R testing when generating primes for use in
RSA Digital Signatures
Parameters

M-R Tests Only

p1 , p 2 , q1 and q 2 > 100 bits

For p1 , p 2 , q1 and q 2 : 28

p and q: 512 bits

For p and q: 5

Error probability = 2 −80
p1 , p 2 , q1 and q 2 > 140 bits

For p1 , p 2 , q1 and q 2 : 38

p and q: 1024 bits

For p and q: 5

Error probability = 2 −112
p1 , p 2 , q1 and q 2 > 170 bits

For p1 , p 2 , q1 and, q 2 : 41

p and q: 1536 bits

For p and q: 4

Error probability = 2 –128
Table C.3. Minimum number of rounds of M-R testing when generating primes for use in
RSA Digital Signatures using an error probability of 2–100
Parameters

M-R Tests Only

p1 , p 2 , q1 and q 2 > 100 bits

For p1 , p 2 , q1 and q 2 : 38

p and q: 512

For p and q: 7

p1 , p 2 , q1 and q 2 > 140 bits

For p1 , p 2 , q1 and q 2 : 32

p and q: 1024 bits

For p and q: 4

p1 , p 2 , q1 and q 2 > 170 bits

For p1 , p 2 , q1 and q 2 : 27

p and q: 1536 bits

For p and q: 3

C.3.1 Miller-Rabin Probabilistic Primality Test
Let RBG be an approved random bit generator.
Input:
1. w

The odd integer to be tested for primality. This will be either p or
q, or one of the auxiliary primes p 1 , p 2 , q 1 or q 2 .

71

2. iterations

The number of iterations of the test to be performed; the value
shall be consistent with Table C.1, C.2 or C.3.

Output:
1. status

The status returned from the validation procedure, where status iseither PROBABLY PRIME or COMPOSITE.

Process:
1. Let a be the largest integer such that 2a divides w−1.
2. m = (w−1) / 2a.
3. wlen = len (w).
4. For i = 1 to iterations do
4.1

Obtain a string b of wlen bits from an RBG.
Comment: Ensure that 1 < b < w−1.

4.2

If ((b ≤ 1) or (b ≥ w−1)), then go to step 4.1.

4.3

z = bm mod w.

4.4

If ((z = 1) or (z = w − 1)), then go to step 4.7.

4.5

For j = 1 to a − 1 do.
4.5.1

z = z2 mod w.

4.5.2

If (z = w−1), then go to step 4.7.

4.5.3

If (z = 1), then go to step 4.6.

4.6 Return COMPOSITE.
4.7 Continue.

Comment: Increment i for the do-loop in
step 4.

5. Return PROBABLY PRIME.
C.3.2 Enhanced Miller-Rabin Probabilistic Primality Test
This method provides additional information when an error is encountered that may be usefulwhen generating or validating RSA moduli. Let RBG be an approved random bit generator.
Input:
1. w

The odd integer to be tested for primality. This will be either p or
q, or one of the auxiliary primes p 1 , p 2 , q 1 or q 2 .

72

2. iterations

The number of iterations of the test to be performed; the value
shall be consistent with Table C.1, C.2 or C.3.

Output:
1. status

The status returned from the validation procedure, where status iseither PROBABLY PRIME, PROVABLY COMPOSITEWITH FACTOR (returned with the factor), and PROVABLYCOMPOSITE AND NOT A POWER OF A PRIME.

Process:
1. Let a be the largest integer such that 2a divides w–1.
2. m = (w–1) / 2a.
3. wlen = len (w).
4. For i = 1 to iterations do
4.1

Obtain a string b of wlen bits from an RBG.
Comment: Ensure that 1 < b < w–1.

4.2

If ((b ≤ 1) or (b ≥ w–1)), then go to step 4.1.

4.3

g = GCD(b, w).

4.4

If (g > 1), then return PROVABLY COMPOSITE WITH FACTOR and thevalue of g.

4.5

z = bm mod w.

4.6

If ((z = 1) or (z = w – 1)), then go to step 4.15.

4.7 For j = 1 to a – 1 do.
Comment: x ≠ 1 and x ≠ w–1.

4.7.1

x = z.

4.7.2

z = x2 mod w.

4.7.3

If (z = w–1), then go to step 4.15.

4.7.4

If (z = 1), then go to step 4.12.

4.8

x = z.

4.9

z = x2 mod w.

Comment: x = b(w–1)/2 mod w and x ≠ w–1.

4.10 If (z = 1), then go to step 4.12.
4.11 x = z.

Comment: x = b(w–1) mod w and x ≠ 1.

73

4.12 g = GCD(x–1, w).
4.13 If (g > 1), then return PROVABLY COMPOSITE WITH FACTOR and thevalue of g.
4.14 Return PROVABLY COMPOSITE AND NOT A POWER OF A PRIME.
4.15 Continue.

Comment: Increment i for the do-loop in
step 4.

5. Return PROBABLY PRIME.
C.3.3 (General) Lucas Probabilistic Primality Test
The following process or its equivalent shall be used as the Lucas test.
Input:C

The candidate odd integer to be tested for primality.

Output:status Where status is either PROBABLY PRIME or COMPOSITE.
Process:
1. Test whether C is a perfect square (see Appendix C.4). If so, return (COMPOSITE).
2. Find the first D in the sequence {5, –7, 9, –11, 13, –15, 17, …} for which the Jacobi
symbol CD = –1. See Appendix C.5 for an approved method to compute the Jacobi()()

Symbol. If CD = 0 for any D in the sequence, return (COMPOSITE).
3. K = C+1.
4. Let K r K r – 1 … K 0 be the binary expansion of K, with K r = 1.
5. Set U r = 1 and V r = 1.
6. For i = r–1 to 0, do
6.1 U temp = U i+1 V i+1 mod C.
6.2 V temp =6.3

Vi +1 2 + DU i +1 2
mod C.
2

If (K i = 1), then

6.3.1

Ui =

Comment: If K i = 1, then do steps 6.3.1 and 6.3.2;
otherwise, do steps 6.3.3 and 6.3.4.
U temp + Vtemp
2

mod C.

74

6.3.2Vi =Vtemp + DU temp
2

mod C.

Else
6.3.3

U i = U temp .

6.3.4

V i = V temp .

7. If (U 0 = 0), then return (PROBABLY PRIME). Otherwise, return (COMPOSITE).
Steps 6.2, 6.3.1 and 6.3.2 contain expressions of the form A/2 mod C, where A is an integer, andC is an odd integer. If A/2 is not an integer (i.e., A is odd), then A/2 mod C may be calculated as(A+C)/2 mod C. Alternatively, A/2 mod C = A·(C+1)/2 mod C, for any integer A, without regard
to A being odd or even.
C.4

Checking for a Perfect Square

The following algorithm may be used to determine whether an n-bit positive integer C is aperfect square:
Input:C The integer to be checked.
Output:status Where status is either PERFECT SQUARE or NOT A PERFECT SQUARE.
Process:
1. Set n, such that 2n > C ≥ 2(n−1).
2. m = n/2.
3. i = 0.
4. Select X 0, such that 2m > X 0 ≥ 2(m−1).
5. Repeat
5.1

i = i + 1.

5.2

X i = ((X i–1 )2 + C)/(2X i–1 ).

Until (X i )2 < 2m + C.
6. If C =  X i  2, thenstatus = PERFECT SQUARE.
Else
status = NOT A PERFECT SQUARE.
75

7. Return status.
Notes:
1. By starting with X 0 > (1/2) Sqrt(C), X 0 − Sqrt(C)is guaranteed to be less than X 0 .
This inequality is maintained in step 5; i.e., X i − Sqrt(C)< X i for all i.
2. For i ≥ 1, 0 ≤ X i − Sqrt(C) = (X i–1 − Sqrt(C))2 / (2 X i–1 ) < X 0 /2i .
In particular, 0 ≤ X m − Sqrt(C) < 1. If Sqrt(C) were an integer, then it would
be equal to the floor of X m .
3. In general, the inequality X i − Sqrt(C) < 1 will occur for values of i that are much lessthan m. To detect this, the fact that 2(m−1) ≤ Sqrt(C) < X i for all i ≥ 1 can be used,
X i − Sqrt(C) = ((X i )2 − C)/( X i + Sqrt(C))≤ ((X i )2 − C)/( 2 Sqrt(C))≤ ((X i )2 − C)/(2m)Thus, the condition (X i )2 < 2m + C implies that X i − Sqrt(C) < 1.
C.5

Jacobi Symbol Algorithm

aThis routine computes the Jacobi symbol   .
nJacobi( ):Input:aAny integer. For this Standard, the initial value is in the sequence {5, –7, 9, –11,13, –15, 17, …}, as determined by Appendix C.3.3.

nAny integer. For this Standard, the initial value is the candidate being tested, as
determined by Appendix C.3.3.

Output:result The calculated Jacobi symbol.
Process:
Comment: a will be in the range 0 ≤ a < n.

1. a = a mod n.
2. If a = 1, or n = 1, then return (1).
3. If a = 0, then return (0).
e

4. Define e and a 1 such that a = 2 a 1 , where a 1 is odd.
76

5. If e is even, then s = 1.
Else if ((n ≡ 1 (mod 8)) or (n ≡ 7 (mod 8))), then s = 1.
Else if ((n ≡ 3 (mod 8)) or (n ≡ 5 (mod 8)), then s = –1.
6. If ((n ≡ 3 (mod 4)) and (a 1 ≡ 3 (mod 4))), then s = –s.
7. n 1 = n mod a 1 .
8. Return (s * Jacobi (n 1 , a 1 )).

Comment: Call this routine recursively.

Example: Compute the Jacobi symbol for a = 5 and n = 3439601197:
1. n is not 1, and a is not 1, so proceed to Step 2.
2. a is not 0, so proceed to Step 3.
3. 5 = 20 * 5, so e = 0, and a 1 = 5.
4. e is even, so s = 1.
5. a 1 is not congruent to 3 mod 4, so do not change s.
6. n 1 = 2 = n mod 5.
7. Compute and return (1 * Jacobi(2, 5)). This calls Jacobi recursively. Compute the Jacobi
symbol for a = 2 and n = 5:
7.1

n is not 1, and a is not 1, so proceed to Step 7.2.

7.2

a is not 0, so proceed to Step 7.3.

7.3

2 = 21 * 1, so e = 1, and a 1 = 1.

7.4

e is odd, and n ≡ 5 (mod 8), so set s = –1.

7.5

n is not 3 mod 4, and a 1 is not 3 mod 4, so proceed to step 7.6.

7.6

n 1 = 0 = n mod 1.

7.7

Return (–1 * Jacobi(0, 1) = –1). This calls Jacobi recursively. Compute the Jacobi
symbol for a = 0 and n = 1:
7.7.1

n = 1, so return 1.

Thus, Jacobi (0,1) = 1, so Jacobi (2,5) = –1*(1) = –1, and Jacobi (5, 3439601197) = 1* (–1) = –1.
C.6

Shawe-Taylor Random_Prime RoutineThis routine is recursive and may be used to construct a provable prime number using a hashfunction.

77

Let Hash( ) be the selected hash function, and let outlen be the bit length of the hash function
output block. The following process or its equivalent shall be used to generate a prime number
for this constructive method.
ST_Random_Prime ( ):Input:
1. length

The length of the prime to be generated.

2. input_seed

The seed to be used for the generation of the requested prime.

Output:1.

status

The status returned from the generation routine, where status iseither SUCCESS or FAILURE. If FAILURE is returned, thenzeros are returned as the other output values.

2. prime

The requested prime.

3

A seed determined during generation.

prime_seed

4. prime_gen_counter(Optional) A counter determined during the generation of theprime.

Process:
1. If (length < 2), then return (FAILURE, 0, 0 {, 0}).
2. If (length ≥ 33), then go to step 14.
3. prime_seed = input_seed.
4. prime_gen_counter = 0.
Comment: Generate a pseudorandom integer
c of length bits.
5. c = Hash(prime_seed) ⊕ Hash(prime_seed + 1).
6. c = 2length – 1 + (c mod 2length – 1).
7. c = (2 ∗ c / 2 ) + 1.
Comment: Set prime to the least odd
integer greater than or equal to c.
8. prime_gen_counter = prime_gen_counter + 1.
9. prime_seed = prime_seed + 2.

78

10. Perform a deterministic primality test on c. For example, since c is small, its primality
can be tested by trial division. See Appendix C.7.
11. If (c is a prime number), then
11.1

prime = c.

11.2

Return (SUCCESS, prime, prime_seed {, prime_gen_counter}).

12. If (prime_gen_counter > (4 ∗ length)), then return (FAILURE, 0, 0 {, 0}).
13. Go to step 5.
14. (status, c 0 , prime_seed, prime_gen_counter) = (ST_Random_Prime (( length / 2+ 1), input_seed).
15. If FAILURE is returned, return (FAILURE, 0, 0 {, 0}).
16. iterations = length / outlen – 1.
17. old_counter = prime_gen_counter.
Comment: Generate a pseudorandom integer
x in the interval [2length – 1, 2length].
18. x = 0.
19. For i = 0 to iterations do
x = x + (Hash(prime_seed + i) ∗ 2i × outlen).
20. prime_seed = prime_seed + iterations + 1.
21. x = 2length – 1 + (x mod 2length – 1).
Comment: Generate a candidate prime c in
the interval [2length – 1, 2length].
22. t = x / (2c 0 ).
23. If (2tc 0 + 1 > 2length), then t = 2length – 1 / (2c 0 ).
24. c = 2tc 0 + 1.
25. prime_gen_counter = prime_gen_counter + 1.
Comment: Test the candidate prime c forprimality; first pick an integer a between 2
and c – 2.
26. a = 0.
27. For i = 0 to iterations doa = a + (Hash(prime_seed + i) ∗ 2 i * outlen).
7928. prime_seed = prime_seed + iterations + 1.
29. a = 2 + (a mod (c – 3)).
30. z = a2t mod c.
31. If ((1 = GCD(z – 1, c)) and (1 = z c0 mod c)), then
31.1

prime = c.

31.2

Return (SUCCESS, prime, prime_seed {, prime_gen_counter}).

32. If (prime_gen_counter ≥ ((4 ∗ length) + old_counter)), then return (FAILURE, 0, 0{, 0}).
33. t = t + 1.
34. Go to step 23.
C.7

Trial Division

An integer is proven to be prime by showing that it has no prime factors less than or equal to its
square root. This procedure is not recommended for testing any integers longer than 10 digits.
To prove that c is prime:
1. Prepare a table of primes less than c . This can be done by applying the sieve procedure in
Appendix C.8.
2. Divide c by every prime in the table. If c is divisible by one of the primes, then declare that c
is composite and exit. If convenient, c may be divided by composite numbers. For example,
rather than preparing a table of primes, it might be more convenient to divide by all integers
except those divisible by 3 or 5.
3. Otherwise, declare that c is prime and exit.
C.8

Sieve Procedure

A sieve procedure is described as follows: Given a sequence of integers Y 0 , Y 0 + 1, … , Y 0 + J, asieve will identify the integers in the sequence that are divisible by primes up to some selected
limit.
Note that the definitions of the mathematical symbols in this process (e.g., h, L, M, p) are
internal to this process only, and should not be confused with their use elsewhere in thisStandard.
Start by selecting a factor base of all the primes p j , from 2 up to some selected limit L. The valueof L is arbitrary and may be determined by computer limitations. A good, typical value of L
would be anywhere from 103 to 105.

80

1. Compute S j =Y 0 mod p j for all p j in the factor base.
2. Initialize an array of length J + 1 to zero.
3. Starting at Y 0 – S j + p j , let every p j th element of the array be set to 1. Do this for theentire length of the array and for every j.
4. When finished, every location in the array that has the value 1 is divisible by some small
prime, and is therefore a composite.
The array can be either a bit array for compactness when memory is small, or a byte array forspeed when memory is readily available. There is no need to sieve the entire sieve interval atonce. The array can be partitioned into suitably small pieces, sieving each piece before going onto the next piece. When finished, every location with the value 0 is a candidate for prime testing.
The amount of work for this procedure is approximately M log log L, where M is the length of
the sieve interval; this is a very efficient procedure for removing composite candidates forprimality testing. If L = 105, the sieve will remove about 96% of all composites.
In some cases, rather than having a set of consecutive integers to sieve, the set of integers to betested consists of integers lying in an arithmetic progression Y 0 , Y 0 + h, Y 0 + 2h, …, Y 0 + Jh,
where h is large and not divisible by any primes in the factor base.
1. Select a factor base and initialize an array of length J + 1 to 0.
2. Compute S j =Y 0 mod p j for all p j in the factor base.
3. Compute T j = h mod p j and r = – S j T j – 1 mod p j .
4. Starting at Y 0 + r, let every p j th element of the array be set to 1. Do this for the entire
length of the array and for every j. Note that the position Y 0 + r in the array actuallydenotes the number Y 0 + rh.
5. When finished, every location in the array that has the value 1 is divisible by some small
prime and is therefore composite.
Note: The prime “2” takes the longest amount of time (M/2) to sieve, since it touches the most
locations in the sieve array. An easy optimization is to combine the initialization of the sieve
array with the sieving of the prime “2”. It is also possible to sieve the prime “3” duringinitialization. These optimizations can save about 1/3 of the total sieve time.
C.9

Compute a Probable Prime Factor Based on Auxiliary Primes

This routine constructs a probable prime (a candidate for p or q) using two auxiliary prime
numbers and the Chinese Remainder Theorem (CRT).
Input:
r 1 and r 2

Two odd prime numbers satisfying
log 2 (r 1 r 2 ) ≤ (nlen/2) – log 2 (nlen/2) – 6.
81

nlen

The desired length of n, the RSA modulus.

e

The public verification exponent.

security_strength

The minimum security strength required for random number
generation.

Output:status

The status returned from the generation procedure, where status iseither SUCCESS or FAILURE. If FAILURE is returned, thenzeros are returned as the other output values.

private_prime_factor The prime factor of n.
X

The random number used during the generation of theprivate_prime_factor.

Process:
1. If (GCD(2r 1 , r 2 ) ≠ 1), then return (FAILURE, 0, 0).
2. R = ((r 2 –1 mod 2r 1 ) * r 2 ) – (((2r 1 )–1 mod r 2 ) * 2r 1 ).
Comment: Apply the CRT, so that R ≡ 1 (mod 2r 1 )and R ≡ –1 (mod r 2 ).
3. Generate a random number X using an approved random number generator that
supports the security_ strength, such that 2 2 nlen / 2−1 ≤ X ≤ 2 nlen / 2 − 1 .

( )(4. Y = X + ((R – X) mod 2r 1 r 2 ).

)

(

)

Comment: Y is the first odd integer ≥ X, such that r 1
is a prime factor of Y–1, and r 2 is a prime factor of
Y+1.
Comment: Determine the requested prime number
by constructing candidates from a sequence andperforming primality tests.

5. i = 0.
6. If (Y ≥ 2nlen/2), then go to step 3.
7. If (GCD(Y–1, e) = 1), then
7.1

Check the primality of Y as specified in Appendix C.3. If PROBABLY PRIME
is not returned, go to step 8.

7.2

private_prime_factor = Y.

7.3

Return (SUCCESS, private_prime_factor, X).

8. i = i + 1.

82

9. If (i ≥ 5(nlen/2)), then return (FAILURE, 0, 0).
10. Y = Y + (2r 1 r 2 ).
11. Go to step 6.
C.10 Construct a Provable Prime (possibly with Conditions), Based onContemporaneously Constructed Auxiliary Provable Primes
The following process (or its equivalent) shall be used to generate an L-bit provable prime p (acandidate for one of the prime factors of an RSA modulus). Note that the use of p in thisspecification is used generically; both RSA prime factors p and q may be generated using thismethod.
If a so-called “strong prime” is required, this process can generate primes p 1 and p 2 (of specified
bit-lengths N 1 and N 2 ) that divide p−1 and p+1, respectively. The resulting prime p will satisfy
the conditions traditionally required of a strong prime, provided that the requested bit-lengths forp 1 and p 2 have appropriate sizes.
Regardless of the bit-lengths selected for p 1 and p 2 , the quantity p−1 will have a prime divisor p 0
whose bit-length is slightly more than half that of p. In addition, the quantity
p 0 −1 will have a prime divisor whose bit-length is slightly more than half that of p 0 .
This algorithm requires that N 1 + N 2 ≤ L – L/2 – 4. Values for N 1 and N 2 should be chosensuch that N 1 + N 2 ≤ (L/2) – log 2 (L) – 7, to ensure that the algorithm can generate as many as 5L
distinct candidates for p.
Let Hash be the selected hash function to be used, and let outlen be the bit length of the hashfunction output block.
Provable_Prime_Construction():Input:
1. L

A positive integer equal to the requested bit-length for p. Note that
acceptable values for L= nlen/2 are computed as specified inAppendix B.3.1, criteria 2(b) and (c), with nlen assuming a value
specified in Table B.1.

2. N 1

A positive integer equal to the requested bit-length for p 1 . If N 1 ≥2, then p 1 is an odd prime of N 1 bits; otherwise, p 1 = 1. Acceptablevalues for N 1 ≥ 2 are provided in Table B.1

3. N 2A positive integer equal to the requested bit-length for p 2 . If N 2 ≥2, then p 2 is an odd prime of N 2 bits; otherwise, p 2 = 1.
Acceptable values for N 2 ≥ 2 are provided in Table B.1

4. firstseed

A bit string equal to the first seed to be used.

83

5. e

The public verification exponent.

Output:
1. status

The status returned from the generation procedure, where status iseither SUCCESS or FAILURE. If FAILURE is returned, thenzeros are returned as the other output values.

2. p, p 1 , p 2

The required prime p, along with p 1 and p 2 having the property
that p 1 divides p−1 and p 2 divides p+1.

3. pseed

A seed determined during generation.

Process:
1. If L, N 1 , and N 2 are not acceptable, then, return (FAILURE, 0, 0, 0, 0).
Comment: Generate p 1 and p 2 , as well as the primep0.
2. If N 1 = 1, then
2.1 p 1 = 1.
2.2 p 2 seed = firstseed.
3. If N 1 ≥ 2, then
3.1 Using N 1 as the length and firstseed as the input_seed, use the random prime
generation routine in Appendix C.6 to obtain p 1 and p 2 seed.
3.2 If FAILURE is returned, then return (FAILURE, 0, 0, 0, 0).
4. If N 2 = 1, then
4.1 p 2 = 1.
4.2 p 0 seed = p 2 seed.
5. If N 2 ≥ 2, then
5.1 Using N 2 as the length and p 2 seed as the input_seed, use the random prime
generation routine in Appendix C.6 to obtain p 2 and p 0 seed.
5.2 If FAILURE is returned, then return (FAILURE, 0, 0, 0, 0).
6. Using L / 2 + 1 as the length and p 0 seed as the input_seed, use the random prime
generation routine in Appendix C.6 to obtain p 0 and pseed. If FAILURE is returned,
then return (FAILURE, 0, 0, 0, 0).
Comment: Generate a (strong) prime p in theinterval [( 2 )(2L−1), 2L −1].
84

7. iterations = L / outlen −1.
8. pgen_counter = 0.
Comment: Generate pseudo-random x in theinterval [( 2 )(2L−1)−1, 2L −1].
9. x = 0.
10. For i = 0 to iterations do
x = x + (Hash(pseed + i))∗ 2 i * outlen.
11. pseed = pseed + iterations + 1.
12. x = ( 2 )(2L−1) + ( x mod (2L − ( 2 )(2L−1) ) ).
Comment: Generate a candidate for the prime p.
13. If (GCD(p 0 p 1 , p 2 ) ≠ 1), then return (FAILURE, 0, 0, 0, 0).
14. Compute y in the interval [1, p 2 ] such that 0 = ( y p 0 p 1 –1) mod p 2 .
15. t = ((2 y p 0 p 1 ) + x)/(2 p 0 p 1 p 2 ).
16. If ((2(t p 2 − y) p 0 p 1 + 1) > 2L), thent = ( (2 y p 0 p 1 ) + ( 2 )(2L−1) ) / (2 p 0 p 1 p 2 ).
Comment: p satisfies
0 = ( p–1) mod (2p 0 p 1 ) and
0 = ( p+1) mod p 2 .
17. p = 2(t p 2 − y) p 0 p 1 + 1.
18. pgen_counter = pgen_counter + 1.
19. If (GCD(p–1, e) = 1), thenComment: Choose an integer a in the interval [2, p–
2].
19.1

a=0

19.2

For i = 0 to iterations doa = a + (Hash(pseed + i))∗ 2 i * outlen.

19.3

pseed = pseed + iterations + 1.

19.4

a = 2 + (a mod (p–3)).
Comment: Test p for primality:

19.5

z = a2(t p2 − y) p1 mod p.
85

19.6

If ((1 = GCD(z–1, p)) and (1 = (z p0 mod p)), then return (SUCCESS, p, p 1 ,p 2 , pseed).

20. If (pgen_counter ≥ 5L), then return (FAILURE, 0, 0, 0, 0).
21. t = t + 1.
22. Go to step 16.

86

Appendix D: Recommended Elliptic Curves for Federal
Government Use
This collection of elliptic curves is recommended for Federal government use and contains
choices for the private key length and underlying fields. These curves were generated using
SHA-1 and the method given in the ANS X9.62 and IEEE Standard 1363-2000 standards. Thisappendix describes the process that was used. Note that these curves are the same as thoseincluded in the previous version of this Standard.
D.1

NIST Recommended Elliptic Curves

D.1.1 Choices
D.1.1.1 Choice of Key Lengths
The principal parameters for elliptic curve cryptography are the elliptic curve E and a designatedpoint G on E called the base point. The base point has order n, which is a large prime. The
number of points on the curve is hn for some integer h (the cofactor), which is not divisible by n.
For efficiency reasons, it is desirable to have the cofactor be as small as possible.
All of the curves given below have cofactors 1, 2, or 4. As a result, the private and public keys
for a curve are approximately the same length.
D.1.1.2 Choice of Underlying Fields
For each key length, two kinds of fields are provided.
•A prime field is the field GF(p), which contains a prime number p of elements. The
elements of this field are the integers modulo p, and the field arithmetic is implemented
in terms of the arithmetic of integers modulo p.

•

A binary field is the field GF(2m), which contains 2m elements for some m (called thedegree of the field). The elements of this field are the bit strings of length m, and the fieldarithmetic is implemented in terms of operations on the bits.

The security strengths for five ranges of the bit length of n is provided in SP 800-57. For thefield GF(p), the security strength is dependent on the length of the binary expansion of p. For thefield GF(2m), the security strength is dependent on the value of m. Table E-1 provides the bit
lengths of the various underlying fields of the curves provided in this appendix. Column 1 lists
the ranges for the bit length of n (also see Table 1 in Section 6.1.1). Column 2 identifies thevalue of p used for the curves over prime fields, where len(p) is the length of the binary
expansion of the integer p. Column 3 provides the value of m for the curves over binary fields.

87

Table D-1: Bit Lengths of the Underlying Fields of the Recommended Curves
Bit Length of n

Prime Field

Binary Field

161 – 223len(p) = 192

m = 163

224 – 255

len(p) = 224

m = 233

256 – 383

len(p) = 256m = 283

384 – 511

len(p) = 384

m = 409≥ 512

len(p) = 521

m = 571

D.1.1.3 Choice of Basis for Binary Fields
To describe the arithmetic of a binary field, it is first necessary to specify how a bit string is to beinterpreted. This is referred to as choosing a basis for the field. There are two common types ofbases: a polynomial basis and a normal basis.
•

A polynomial basis is specified by an irreducible polynomial modulo 2, called the fieldpolynomial. The bit string (a m–1 … a 2 a 1 a 0 ) is taken to represent the polynomial
a m–1 t m–1 + …+ a 2 t2 + a 1 t + a 0
over GF(2). The field arithmetic is implemented as polynomial arithmetic modulo p(t),where p(t) is the field polynomial.

•

A normal basis is specified by an element θ of a particular kind. The bit string (a 0 a 1 a 2… a m–1 ) is taken to represent the elementa 0 θ + a 1 θ 2 + a 2 θ 2 + … + a m–1 θ 2
2

m–1

.

Normal basis field arithmetic is not easy to describe or efficient to implement in general,
except for a special class called Type T low-complexity normal bases. For a given field
degree m, the choice of T specifies the basis and the field arithmetic (see Appendix D.3).
There are many polynomial bases and normal bases from which to choose. The following
procedures are commonly used to select a basis representation.
•

Polynomial Basis: If an irreducible trinomial tm + tk + 1 exists over GF (2), then the fieldpolynomial p(t) is chosen to be the irreducible trinomial with the lowest-degree middle
term tk . If no irreducible trinomial exists, then a pentanomial t m + t a + t b + t c + 1 isselected. The particular pentanomial chosen has the following properties: the second termta has the lowest degree m; the third term tb has the lowest degree among all irreducible
pentanomials of degree m and second term ta; and the fourth term tc has the lowest degree
among all irreducible pentanomials of degree m, second term ta, and third term tb.

88•

Normal Basis: Choose the Type T low-complexity normal basis with the smallest T.

For each binary field, the parameters are given for the above basis representations.
D.1.1.4 Choice of Curves
Two kinds of curves are given:•

Pseudo-random curves are those whose coefficients are generated from the output of aseeded cryptographic hash function. If the domain parameter seed value is given alongwith the coefficients, it can be easily verified that the coefficients were generated by thatmethod.

•Special curves are those whose coefficients and underlying field have been selected to
optimize the efficiency of the elliptic curve operations.

For each curve size range, the following curves are given:→ A pseudo-random curve over GF(p).
→ A pseudo-random curve over GF(2m).
→ A special curve over GF(2m) called a Koblitz curve or anomalous binary curve.
The pseudo-random curves were generated as specified in ANS X9.62 using SHA-1.
D.1.1.5 Choice of Base Points
Any point of order n can serve as the base point. Each curve is supplied with a sample basepoint G = (G x , G y ). Users may want to generate their own base points to ensure cryptographic
separation of networks. See ANS X9.62 or IEEE Standard 1363-2000.
D.1.2 Curves over Prime Fields
For each prime p, a pseudo-random curve
E : y2 ≡ x3 – 3x +b (mod p)of prime order n is listed 4. (Thus, for these curves, the cofactor is always h = 1.) The following
parameters are given:

4•

The prime modulus p•

The order n•

The 160-bit input seed SEED to the SHA-1 based algorithm (i.e., the domain parameter
seed)

•

The output c of the SHA-1 based algorithm

The selection a ≡ -3 for the coefficient of x was made for reasons of efficiency; see IEEE Std 1363-2000.

89

•The coefficient b (satisfying b2 c ≡ –27 (mod p))•

The base point x coordinate G x•

The base point y coordinate G y

The integers p and n are given in decimal form; bit strings and field elements are given inhexadecimal.
D.1.2.1 Curve P-192
p=6277101735386680763835789423207666416083908700390324961279

n=6277101735386680763835789423176059013767194773182842284081

SEED = 3045ae6f c8422f64 ed579528 d38120ea e12196d5c=3099d2bb bfcb2538 542dcd5f b078b6ef 5f3d6fe2 c745de65b=64210519 e59c80e7 0fa7e9ab 72243049 feb8deec c146b9b1

Gx=

188da80e b03090f6 7cbf20eb 43a18800 f4ff0afd 82ff1012

Gy=

07192b95 ffc8da78 631011ed 6b24cdd5 73f977a1 1e794811

D.1.2.2 Curve P-224
p=2695994666715063979466701508701963067355791626002630814351
0066298881

n=2695994666715063979466701508701962594045780771442439172168
2722368061

SEED = bd713447 99d5c7fc dc45b59f a3b9ab8f 6a948bc5c=5b056c7e 11dd68f4 0469ee7f 3c7a7d74 f7d12111 6506d031
218291fbb=b4050a85 0c04b3ab f5413256 5044b0b7 d7bfd8ba 270b3943
2355ffb4

Gx=

b70e0cbd 6bb4bf7f 321390b9 4a03c1d3 56c21122 343280d6
115c1d21

Gy=bd376388 b5f723fb 4c22dfe6 cd4375a0 5a074764 44d58199
85007e3490D.1.2.3 Curve P-256
p=1157920892103562487626974469494075735300861434152903141955
33631308867097853951

n=115792089210356248762697446949407573529996955224135760342
422259061068512044369

SEED = c49d3608 86e70493 6a6678e1 139d26b7 819f7e90
c=

7efba166 2985be94 03cb055c 75d4f7e0 ce8d84a9 c5114abc
af317768 0104fa0db=5ac635d8 aa3a93e7 b3ebbd55 769886bc 651d06b0 cc53b0f6
3bce3c3e 27d2604bGx=

6b17d1f2 e12c4247 f8bce6e5 63a440f2 77037d81 2deb33a0
f4a13945 d898c296

Gy=4fe342e2 fe1a7f9b 8ee7eb4a 7c0f9e16 2bce3357 6b315ece
cbb64068 37bf51f5

D.1.2.4 Curve P-384
p=3940200619639447921227904010014361380507973927046544666794
8293404245721771496870329047266088258938001861606973112319

n=3940200619639447921227904010014361380507973927046544666794
6905279627659399113263569398956308152294913554433653942643

SEED = a335926a a319a27a 1d00896a 6773a482 7acdac73c=

79d1e655 f868f02f ff48dcde e14151dd b80643c1 406d0ca1
0dfe6fc5 2009540a 495e8042 ea5f744f 6e184667 cc722483

b=b3312fa7 e23ee7e4 988e056b e3f82d19 181d9c6e fe814112
0314088f 5013875a c656398d 8a2ed19d 2a85c8ed d3ec2aef

Gx=aa87ca22 be8b0537 8eb1c71e f320ad74 6e1d3b62 8ba79b98
59f741e0 82542a38 5502f25d bf55296c 3a545e38 72760ab7

G y=3617de4a 96262c6f 5d9e98bf 9292dc29 f8f41dbd 289a147c
e9da3113 b5f0b8c0 0a60b1ce 1d7e819d 7a431d7c 90ea0e5f
91

D.1.2.5 Curve P-521
p=686479766013060971498190079908139321726943530014330540939
446345918554318339765605212255964066145455497729631139148
0858037121987999716643812574028291115057151

n=686479766013060971498190079908139321726943530014330540939
446345918554318339765539424505774633321719753296399637136
3321113864768612440380340372808892707005449

SEED = d09e8800 291cb853 96cc6717 393284aa a0da64ba
c=0b4 8bfa5f42 0a349495 39d2bdfc 264eeeeb 077688e4
4fbf0ad8 f6d0edb3 7bd6b533 28100051 8e19f1b9 ffbe0fe9
ed8a3c22 00b8f875 e523868c 70c1e5bf 55bad637b=051 953eb961 8e1c9a1f 929a21a0 b68540ee a2da725b
99b315f3 b8b48991 8ef109e1 56193951 ec7e937b 1652c0bd
3bb1bf07 3573df88 3d2c34f1 ef451fd4 6b503f00

Gx=

c6 858e06b7 0404e9cd 9e3ecb66 2395b442 9c648139
053fb521 f828af60 6b4d3dba a14b5e77 efe75928 fe1dc127
a2ffa8de 3348b3c1 856a429b f97e7e31 c2e5bd66

Gy=

118 39296a78 9a3bc004 5c8a5fb4 2c7d1bd9 98f54449
579b4468 17afbd17 273e662c 97ee7299 5ef42640 c550b901
3fad0761 353c7086 a272c240 88be9476 9fd16650

D.1.3 Curves over Binary Fields
For each field degree m, a pseudo-random curve is given, along with a Koblitz curve. The
pseudo-random curve has the form
E: y 2 + x y = x 3 + x 2 + b,
and the Koblitz curve has the form
E a : y2 + x y = x 3 + ax 2 + 1,
where a = 0 or 1.
For each pseudorandom curve, the cofactor is h = 2. The cofactor of each Koblitz curve is h = 2
if a = 1, and h = 4 if a = 0.

92

The coefficients of the pseudo-random curves, and the coordinates of the base points of bothkinds of curves, are given in terms of both the polynomial and normal basis representations
discussed in Appendix D.1.1.3.
For each m, the following parameters are given:
Field Representation:•

The normal basis type T•

The field polynomial (a trinomial or pentanomial)

Koblitz Curve:•

The coefficient a•

The base point order n•

The base point x coordinate G x•

The base point y coordinate G yPseudo-random curve:•

The base point order nPseudo-random curve (Polynomial Basis representation):•

The coefficient b•

The base point x coordinate G x•

The base point y coordinate G yPseudo-random curve (Normal Basis representation):•

The 160-bit input seed SEED to the SHA-1 based algorithm (i.e., the domain parameter
seed)

•

The coefficient b (i.e., the output of the SHA-1 based algorithm)•

The base point x coordinate G x•

The base point y coordinate G yIntegers (such as T, m, and n) are given in decimal form; bit strings and field elements are given
in hexadecimal.
D.1.3.1 Degree 163 Binary Field
T =4p(t) = t 163 + t 7 + t 6 + t 3 + 1

93D.1.3.1.1 Curve K-163a=1n=5846006549323611672814741753598448348329118574063

Polynomial Basis:G x = 2 fe13c053 7bbc11ac aa07d793 de4e6d5e 5c94eee8
G y = 2 89070fb0 5d38ff58 321f2e80 0536d538 ccdaa3d9
Normal Basis:G x = 0 5679b353 caa46825 fea2d371 3ba450da 0c2a4541
G y = 2 35b7c671 00506899 06bac3d9 dec76a83 5591edb2
D.1.3.1.2 Curve B-163
n=5846006549323611672814742442876390689256843201587

Polynomial Basis:b=

2 0a601907 b8c953ca 1481eb10 512f7874 4a3205fd

G x = 3 f0eba162 86a2d57e a0991168 d4994637 e8343e36
G y = 0 d51fbc6c 71a0094f a2cdd545 b11c5c0c 797324f1
Normal Basis:
SEED =

85e25bfe 5c86226c db12016f 7553f9d0 e693a268

b=

6 645f3cac f1638e13 9c6cd13e f61734fb c9e3d9fb

Gx=

0 311103c1 7167564a ce77ccb0 9c681f88 6ba54ee8

Gy=

3 33ac13c6 447f2e67 613bf700 9daf98c8 7bb50c7f

D.1.3.2 Degree 233 Binary Field
T =2p(t) =

t 233 + t 74 + 1

94

D.1.3.2.1 Curve K-233
a=

0

n=

345087317339528189371737793113851276057094098886225212\
6328087024741343

Polynomial Basis:Gx=

172 32ba853a 7e731af1 29f22ff4 149563a4 19c26bf5
0a4c9d6e efad6126

Gy=

1db 537dece8 19b7f70f 555a67c4 27a8cd9b f18aeb9b
56e0c110 56fae6a3

Normal Basis:Gx=

0fd e76d9dcd 26e643ac 26f1aa90 1aa12978 4b71fc07
22b2d056 14d650b3

Gy=

064 3e317633 155c9e04 47ba8020 a3c43177 450ee036
d6335014 34cac978D.1.3.2.2 Curve B-233
n=690174634679056378743475586227702555583981273734501355\
5379383634485463

Polynomial Basis:b=

066 647ede6c 332c7f8c 0923bb58 213b333b 20e9ce42
81fe115f 7d8f90ad

Gx=

0fa c9dfcbac 8313bb21 39f1bb75 5fef65bc 391f8b36
f8f8eb73 71fd558b

Gy=

100 6a08a419 03350678 e58528be bf8a0bef f867a7ca
36716f7e 01f81052

95

Normal Basis:
SEED =

74d59ff0 7f6b413d 0ea14b34 4b20a2db 049b50c3

b=1a0 03e0962d 4f9a8e40 7c904a95 38163adb 82521260
0c7752ad 52233279

Gx=

18b 863524b3 cdfefb94 f2784e0b 116faac5 4404bc91
62a363ba b84a14c5

Gy=049 25df77bd 8b8ff1a5 ff519417 822bfedf 2bbd7526
44292c98 c7af6e02

D.1.3.3 Degree 283 Binary Field
T =6p(t) = t 283 + t 12 + t 7 + t 5 + 1
D.1.3.3.1 Curve K-283
a=

0

n=

3885337784451458141838923813647037813284811733793061324
295874997529815829704422603873

Polynomial Basis:Gx=503213f 78ca4488 3f1a3b81 62f188e5 53cd265f 23c1567a
16876913 b0c2ac24 58492836

Gy=

1ccda38 0f1c9e31 8d90f95d 07e5426f e87e45c0 e8184698
e4596236 4e341161 77dd2259

Normal Basis:Gx=

3ab9593 f8db09fc 188f1d7c 4ac9fcc3 e57fcd3b db15024b
212c7022 9de5fcd9 2eb0ea60

Gy=2118c47 55e7345c d8f603ef 93b98b10 6fe8854f feb9a3b3
04634cc8 3a0e759f 0c2686b1

96

D.1.3.3.2 Curve B-283
n=7770675568902916283677847627294075626569625924376904889
109196526770044277787378692871
Polynomial Basis:b=

27b680a c8b8596d a5a4af8a 19a0303f ca97fd76 45309fa2
a581485a f6263e31 3b79a2f5Gx=

5f93925 8db7dd90 e1934f8c 70b0dfec 2eed25b8 557eac9c
80e2e198 f8cdbecd 86b12053

Gy=

3676854 fe24141c b98fe6d4 b20d02b4 516ff702 350eddb0826779c8 13f0df45 be8112f4

Normal Basis:
SEED =
b=77e2b073 70eb0f83 2a6dd5b6 2dfc88cd 06bb84be
157261b 894739fb 5a13503f 55f0b3f1 0c560116 66331022
01138cc1 80c0206b dafbc951

Gx=749468e 464ee468 634b21f7 f61cb700 701817e6 bc36a236
4cb8906e 940948ea a463c35d

Gy=62968bd 3b489ac5 c9b859da 68475c31 5bafcdc4 ccd0dc90
5b70f624 46f49c05 2f49c08c

D.1.3.4 Degree 409 Binary Field
T =4p(t) = t 409 + t 87 + 1
D.1.3.4.1 Curve K-409
a=0n=33052798439512429947595765401638551991420234148214060964\
232439502288071128924919105067325845777745801409636659061
7731358671

97

Polynomial Basis:Gx=

060f05f 658f49c1 ad3ab189 0f718421 0efd0987 e307c84c
27accfb8 f9f67cc2 c460189e b5aaaa62 ee222eb1 b35540cf
e9023746

G y = 1e36905 0b7c4e42 acba1dac bf04299c 3460782f 918ea427
e6325165 e9ea10e3 da5f6c42 e9c55215 aa9ca27a 5863ec48
d8e0286b
Normal Basis:Gx=1b559c7 cba2422e 3affe133 43e808b5 5e012d72 6ca0b7e6
a63aeafb c1e3a98e 10ca0fcf 98350c3b 7f89a975 4a8e1dc0
713cec4aGy=

16d8c42 052f07e7 713e7490 eff318ba 1abd6fef 8a5433c8
94b24f5c 817aeb79 852496fb ee803a47 bc8a2038 78ebf1c4
99afd7d6

D.1.3.4.2 Curve B-409
n=6610559687902485989519153080327710398284046829642812192
84648798304157774827374805208143723762179110965979867288
366567526771

Polynomial Basis:b=

021a5c2 c8ee9feb 5c4b9a75 3b7b476b 7fd6422e f1f3dd67
4761fa99 d6ac27c8 a9a197b2 72822f6c d57a55aa 4f50ae31
7b13545f

Gx=

15d4860 d088ddb3 496b0c60 64756260 441cde4a f1771d4d
b01ffe5b 34e59703 dc255a86 8a118051 5603aeab 60794e54
bb7996a7

98

Gy=061b1cf ab6be5f3 2bbfa783 24ed106a 7636b9c5 a7bd198d
0158aa4f 5488d08f 38514f1f df4b4f40 d2181b36 81c364ba
0273c706

Normal Basis:
SEED =

4099b5a4 57f9d69f 79213d09 4c4bcd4d 4262210bb=

124d065 1c3d3772 f7f5a1fe 6e715559 e2129bdf a04d52f7
b6ac7c53 2cf0ed06 f610072d 88ad2fdc c50c6fde 72843670
f8b3742aGx=0ceacbc 9f475767 d8e69f3b 5dfab398 13685262 bcacf22b
84c7b6dd 981899e7 318c96f0 761f77c6 02c016ce d7c548de
830d708f

Gy=

199d64b a8f089c6 db0e0b61 e80bb959 34afd0ca f2e8be76
d1c5e9af fc7476df 49142691 ad303902 88aa09bc c59c1573
aa3c009a

D.1.3.5 Degree 571 Binary Field
T =10p(t) = t 571 + t 10 + t 5 + t 2 + 1
D.1.3.5.1 Curve K-571
a=0n=

1932268761508629172347675945465993672149463664853217499
32861762572575957114478021226813397852270671183470671280
08253514612736749740666173119296824216170925035557336852
76673

Polynomial Basis:Gx=

26eb7a8 59923fbc 82189631 f8103fe4 ac9ca297 0012d5d4
60248048 01841ca4 43709584 93b205e6 47da304d b4ceb08c
99

bbd1ba39 494776fb 988b4717 4dca88c7 e2945283 a01c8972
Gy=349dc80 7f4fbf37 4f4aeade 3bca9531 4dd58cec 9f307a54
ffc61efc 006d8a2c 9d4979c0 ac44aea7 4fbebbb9 f772aedc
b620b01a 7ba7af1b 320430c8 591984f6 01cd4c14 3ef1c7a3

Normal Basis:Gx=

04bb2db a418d0db 107adae0 03427e5d 7cc139ac b465e593
4f0bea2a b2f3622b c29b3d5b 9aa7a1fd fd5d8be6 6057c100
8e71e484 bcd98f22 bf847642 37673674 29ef2ec5 bc3ebcf7

Gy=

44cbb57 de20788d 2c952d7b 56cf39bd 3e89b189 84bd124e
751ceff4 369dd8da c6a59e6e 745df44d 8220ce22 aa2c852c
fcbbef49 ebaa98bd 2483e331 80e04286 feaa2530 50caff60

D.1.3.5.2 Curve B-571
n=3864537523017258344695351890931987344298927329706434998
65723525145151914228956042453614399938941577308313388112
19269444862468724628168130702345282883033324113931911052
85703

Polynomial Basis:b=

2f40e7e 2221f295 de297117 b7f3d62f 5c6a97ff cb8ceff1
cd6ba8ce 4a9a18ad 84ffabbd 8efa5933 2be7ad67 56a66e29
4afd185a 78ff12aa 520e4de7 39baca0c 7ffeff7f 2955727a

G x=

303001d 34b85629 6c16c0d4 0d3cd775 0a93d1d2 955fa80aa5f40fc8 db7b2abd bde53950 f4c0d293 cdd711a3 5b67fb14
99ae6003 8614f139 4abfa3b4 c850d927 e1e7769c 8eec2d19

G y=

37bf273 42da639b 6dccfffe b73d69d7 8c6c27a6 009cbbca
1980f853 3921e8a6 84423e43 bab08a57 6291af8f 461bb2a8
b3531d2f 0485c19b 16e2f151 6e23dd3c 1a4827af 1b8ac15b100

Normal Basis:
SEED =

2aa058f7 3a0e33ab 486b0f61 0410c53a 7f132310

b=3762d0d 47116006 179da356 88eeaccf 591a5cde a7500011
8d9608c5 9132d434 26101a1d fb377411 5f586623 f75f0000
1ce61198 3c1275fa 31f5bc9f 4be1a0f4 67f01ca8 85c74777

Gx=

0735e03 5def5925 cc33173e b2a8ce77 67522b46 6d278b65
0a291612 7dfea9d2 d361089f 0a7a0247 a184e1c7 0d417866
e0fe0feb 0ff8f2f3 f9176418 f97d117e 624e2015 df1662a8

G y=

04a3642 0572616c df7e606f ccadaecf c3b76dab 0eb1248d
d03fbdfc 9cd3242c 4726be57 9855e812 de7ec5c5 00b4576a
24628048 b6a72d88 0062eed0 dd34b109 6d3acbb6 b01a4a97

D.2

Implementation of Modular Arithmetic

The prime moduli in the above examples are of a special type (called generalized Mersenne
numbers) for which modular multiplication can be carried out more efficiently than in general.
This section provides the rules for implementing this faster arithmetic for each of the prime
moduli appearing in the examples.
The usual way to multiply two integers (mod m) is to take the integer product and reduce it (modm). One therefore has the following problem: given an integer A less than m 2, compute
B = A mod m.
In general, one must obtain B as the remainder of an integer division. If m is a generalized
Mersenne number, however, then B can be expressed as a sum or difference (mod m) of a small
number of terms. To compute this expression, the integer sum or difference can be evaluated andthe result reduced modulo m. The latter reduction can be accomplished by adding or subtracting
a few copies of m.
The prime modulus p for each of the five example curves is a generalized Mersenne number.
D.2.1 Curve P-192
The modulus for this curve is p = 2 192 – 2 64 – 1. Every integer A less than p2 can be written as
A = A5 ⋅ 2 320 + A4 ⋅ 2 256 + A3 ⋅ 2192 + A2 ⋅ 2128 + A1 ⋅ 2 64 + A0 ,where each A i is a 64-bit integer. As a concatenation of 64-bit words, this can be denoted by
A = (A 5 || A 4 || A 3 || A 2 || A 0 ).
The expression for B is
B = T + S 1 + S 2 + S 3 mod p,101

where the 192-bit terms are given by
T =(A 2 || A 1 || A 0 )S 1 = (A 3 || A 3 )S 2 = (A 4 || A 4 || 0)
S 3 = (A 5 || A 5 || A 5 ).
D.2.2 Curve P-224
The modulus for this curve is p = 2 224 − 2 96 + 1. Every integer A less than p2 can be written as:
A = A13 ⋅ 2 416 + A12 ⋅ 2384 + A11 ⋅ 2352 + A10 ⋅ 2320 + A9 ⋅ 2 288 + A8 ⋅ 2 256 + A7 ⋅ 2 224 + A6 ⋅ 2192 +A5 ⋅ 2160 + A4 ⋅ 2128 + A3 ⋅ 296 + A2 ⋅ 2 64 + A1 ⋅ 232 + A0 ,where each A i is a 32-bit integer. As a concatenation of 32-bit words, this can be denoted by:
A = ( A 13 || A 12 || … || A 0 ).
The expression for B is:
B = T + S 1 + S 2 – D 1 – D 2 mod p,
where the 224-bit terms are given by:T =( A 6 || A 5 || A 4 || A 3 || A 2 || A 1 || A 0 )S1 =( A 10 || A 9 || A 8 || A 7 || 0 || 0 || 0 )S2 =( 0 || A 13 || A 12 || A 11 || 0 || 0 || 0 )D1 =( A 13 || A 12 || A 11 || A 10 || A 9 || A 8 || A 7 )D2 =( 0 || 0 || 0 || 0 || A 13 || A 12 || A 11 ).

D.2.3 Curve P-256
The modulus for this curve is p = 2256 – 2224 + 2192 + 296 – 1. Every integer A less than p2 can be
written as:
A = A15 ⋅ 2 480 + A14 ⋅ 2 448 + A13 ⋅ 2 416 + A12 ⋅ 2384 + A11 ⋅ 2352 + A10 ⋅ 2320 + A9 ⋅ 2 288 + A8 ⋅ 2 256 +A7 ⋅ 2 224 + A6 ⋅ 2192 + A5 ⋅ 2160 + A4 ⋅ 2128 + A3 ⋅ 296 + A2 ⋅ 2 64 + A1 ⋅ 232 + A0 ,where each A i is a 32-bit integer. As a concatenation of 32-bit words, this can be denoted by
A = (A 15 || A 14 || ⋅ ⋅ ⋅ || A 0 ).
The expression for B is:102

B = T + 2S 1 + 2S 2 + S 3 + S 4 – D 1 – D 2 – D 3 – D 4 mod p,
where the 256-bit terms are given by:
T = ( A 7 || A 6 || A 5  || A 4 || A 3 || A 2 || A 1 || A 0 )S 1 = ( A 15 || A 14  || A 13 || A 12 || A 11 || 0 || 0 || 0 )S 2 = ( 0 || A 15 || A 14  || A 13 || A 12 || 0 || 0 || 0 )S 3 = ( A 15 || A 14  || 0 || 0 || 0 || A 10 || A 9 || A 8 )
S 4 = ( A 8 || A 13 || A 15 || A 14 || A 13 || A 11 || A 10 || A 9 )
D 1 = ( A 10 || A 8 || 0 || 0 || 0 || A 13 || A 12 || A 11 )D 2 = ( A 11 || A 9 || 0 || 0 || A 15 || A 14 || A 13 || A 12  )D 3 = ( A 12  || 0 || A 10 || A 9 || A 8 || A 15 || A 14 || A 13 )D 4 = ( A 13 || 0 || A 11 || A 10 || A 9 || 0 || A 15 || A 14 )D.2.4 Curve P-384
The modulus for this curve is p = 2 384 – 2 128 – 2 96 + 2 32 – 1. Every integer A less than p2 canbe written as:
A = A23 ⋅ 2 736 + A22 ⋅ 2 704 + A21 ⋅ 2 672 + A20 ⋅ 2 640 + A19 ⋅ 2 608 + A18 ⋅ 2576 + A17 ⋅ 2544 + A16 ⋅ 2512 +A15 ⋅ 2 480 + A14 ⋅ 2 448 + A13 ⋅ 2 416 + A12 ⋅ 2384 + A11 ⋅ 2352 + A10 ⋅ 2320 + A9 ⋅ 2 288 + A8 ⋅ 2 256 +A7 ⋅ 2 224 + A6 ⋅ 2192 + A5 ⋅ 2160 + A4 ⋅ 2128 + A3 ⋅ 296 + A2 ⋅ 2 64 + A1 ⋅ 232 + A0 ,where each A i is a 32-bit integer. As a concatenation of 32-bit words, this can be denoted by
A = (A 23 || A 22 || ⋅ ⋅ ⋅ || A 0 ).
The expression for B is:
B = T + 2S 1 + S 2 + S 3 + S 4 + S 5 + S 6 – D 1 – D 2 – D 3 mod p,
where the 384-bit terms are given by:
T =(A 11 || A 10 || A 9 || A 8 || A 7 || A 6 || A 5 || A 4 || A 3 || A 2 || A 1 || A 0 )S1 =( 0 || 0 || 0 || 0 || 0 || A 23 || A 22 || A 21 || 0 || 0 || 0 || 0 )S2 =(A 23 || A 22 || A 21 || A 20 || A 19 || A 18 || A 17 || A 16 || A 15 || A 14 || A 13 || A 12 )S3 =(A 20 || A 19 || A 18 || A 17 || A 16 || A 15 || A 14 || A 13 || A 12 || A 23 || A 22 || A 21 )S4 =( A 19 || A 18 || A 17 || A 16 || A 15 || A 14 || A 13 || A 12 || A 20 || 0 || A 23 || 0 )S5 =( 0 || 0 || 0 || 0 || A 23 || A 22 || A 21 || A 20 || 0 || 0 || 0 || 0 )

103

S6 =

( 0 || 0 || 0 || 0 || 0 || 0 || A 23 || A 22 || A 21 || 0 || 0 || A 20 )D1 =(A 22 || A 21 || A 20 || A 19 || A 18 || A 17 || A 16 || A 15 || A 14 || A 13 || A 12 || A 23 )D2 =( 0 || 0 || 0 || 0 || 0 || 0 || 0 || A 23 || A 22 || A 21 || A 20 || 0 )D3 =( 0 || 0 || 0 || 0 || 0 || 0 || 0 || A 23 || A 23 || 0 || 0 || 0 ).

D.2.5 Curve P-521
The modulus for this curve is p = 2 521 – 1. Every integer A less than p2 can be written
A = A 1 ⋅ 2521 + A 0 ,where each A i is a 521-bit integer. As a concatenation of 521-bit words, this can be denoted by
A = (A 1 || A 0 ).
The expression for B is:
B = (A 0 + A 1 ) mod p.
D.3

Normal Bases

The elements of GF(2m) are expressed in terms of the type T normal basis 5 B for GF(2m), forsome T. Each element has a unique representation as a bit string:( a 0 a 1 … a m–1 ).
The arithmetic operations are performed as follows.
Addition: addition of two elements is implemented by bit-wise addition modulo 2. Thus, forexample,(1100111) + (1010010) = (0110101).
Squaring: if

α = ( a 0 a 1 … a m–1 )then

α2 = (a m–1 a 0 a 1 … a m–2 ).
Multiplication: to perform multiplication, a function F(u,v) is constructed on inputs

5

It is assumed in this section that m is odd and T is even, since this is the only case considered in this Standard.

104u = ( u 0 u 1 … u m–1 )andv = ( v 0 v 1 … v m–1 )as follows.
1. Set p ← Tm + 1.
2. Let u be an integer having order T modulo p.
3. Compute the sequence F (1), F (2), … ,F (p–1) as follows:
3.1

Set w← 1.

3.2

For j from 0 to T–1 do
3.2.1

Set n ← w.

3.2.2

For i = 0 to m–1 do

3.2.3

3.2.2.1

Set F(n) ← i.

3.2.2.2

Set n ← 2n mod p.

Set w ← uw mod p.

4. Output the formula:p −2

F (u , v) := ∑ u F ( k +1) v F ( p − k ) .
k =1

This computation need only be performed once per basis.
Given the function F for B, the product( c 0 c 1 … c m–1 ) = ( a 0 a 1 … a m–1 ) * ( b 0 b 1 … b m–1 )is computed as follows:
1. Set ( u 0 u 1 … u m–1 ) ← ( a 0 a 1 . . . a m–1 ).
2. Set ( v 0 v 1 … v m–1 ) ← ( b 0 b 1 . . . b m–1 ).
3. For k = 0 to m – 1 do
3.1

Computec k = F(u, v).

3.2

Set u ← LeftShift (u) and v ← LeftShift (v), where LeftShift denotes the circularleft shift operation.

4. Output c = ( c 0 c 1 … c m–1 ).
105

Example: For the type 4 normal basis for GF(27), p = 29 and u = 12 or 17. Thus, the values of F
are given by:
F (1) = 0 F (8) = 3 F (15) = 6 F (22) = 5
F (2) = 1 F (9) = 3 F (16) = 4 F (23) = 6
F (3) = 5 F (10) = 2 F (17) = 0 F (24) = 1
F (4) = 2 F (11) = 4 F (18) = 4 F (25) = 2
F (5) = 1 F (12) = 0 F (19) = 2 F (26) = 5
F (6) = 6 F (13) = 4 F (20) = 3 F (27) = 1
F (7) = 5 F (14) = 6 F (21) = 3 F (28) = 0
Therefore,F (u, v) = u 0 v 1 + u 1 (v 0 + v 2 + v 5 + v 6 ) + u 2 (v 1 + v 3 + v 4 + v 5 ) + u 3 (v 2 + v 5 ) +u 4 (v 2 + v 6 ) + u 5 (v 1 + v 2 + v 3 + v 6 ) + u 6 (v 1 + v 4 + v 5 + v 6 ).
Thus, ifa = (1 0 1 0 1 1 1) and b = (1 1 0 0 0 0 1),
thenc 0 = F ( (1 0 1 0 1 1 1), (1 1 0 0 0 0 1) ) = 1,c 1 = F ( (0 1 0 1 1 1 1), (1 0 0 0 0 1 1) ) = 0,c 6 = F ( (1 1 0 1 0 1 1), (1 1 1 0 0 0 0) ) = 1,
so that c = ab = (1 0 1 1 0 0 1).
D.4

Scalar Multiplication on Koblitz Curves

This section describes a particularly efficient method of computing the scalar multiple nP on the
Koblitz curve E a over GF(2m).
The operation τ is defined by:τ (x, y) = (x2, y2).
When the normal basis representation is used, then the operation τ is implemented by performing
right circular shifts on the bit strings representing x and y.
Given m and a, define the following parameters:•

C is some integer greater than 5.

106

•

µ = (–1)1–a.

•

For i = 0 and i = 1, define the sequence s i (m) by:s i (0) = 0, s i (1) = 1 – i,s i (m) = µ • s i (m – 1) – 2 s i (m – 2) + (–1)i•Define the sequence V(m)V(0) = 2, V(1) = µ
V(m) = µ • v(m –1) – 2V(m – 2).

For the example curves, the quantities s i (m) and V(m) are as follows.
Curve K-163:
s 0 (163) = 2579386439110731650419537
s 1 (163) = –755360064476226375461594
V(163) = –4845466632539410776804317
Curve K-233:
s 0 (233) = –27859711741434429761757834964435883
s 1 (233) = –44192136247082304936052160908934886
V(233) = –137381546011108235394987299651366779
Curve K-283:
s 0 (283) = –665981532109049041108795536001591469280025
s 1 (283) = 1155860054909136775192281072591609913945968
V(283) = 7777244870872830999287791970962823977569917
Curve K-409:
s 0 (409) = –18307510456002382137810317198756461378590542487556869338419259
s 1 (409) = –8893048526138304097196653241844212679626566100996606444816790
V(409)=

10457288737315625927447685387048320737638796957687575791173829

Curve K-571:
s 0 (571) = –3737319446876463692429385892476115567147293964596131024123406420\
235241916729983261305
s 1 (571) = –3191857706446416099583814595948959674131968912148564658610565117\
58982848515832612248752

107

V(571)=–1483809269816914138996191402970514903645425741804939362329123395\
34208516828973111459843

The following algorithm computes the scalar multiple nP on the Koblitz curve E a over GF(2m).
The average number of elliptic additions and subtractions is at most ∼ 1 + (m/3), and is at most ∼m/3 with probability at least 1 – 25–C.
1. For i = 0 to 1 do
1.1n′ ←  n / 2a–C + (m–9) / 2.

1.2

g′ ← s i (m) · n′.

1.3h′ ←  g′ / 2m .

1.4

j′ ← V(m) · h′.

1.5l′ ← Round((g′ + j′) / 2(m+5) / 2).

1.6

λ i ← l′ / 2C.

1.7

f i ← Round(λ i ).

1.8

η i ← λ i – f i. .

1.9

h i ← 0.

2. η ← 2 η 0 + µ η 1 .
3. If (η ≥ 1),then
if (η o – 3 µη 1 < –1)
then set h 1 ← µelse set h 0 ← 1.
else
if (η 0 + 4 µ η 1 ≥ 2)then set h 1 ← µ.
4. If (η < –1)
thenif (η 0 – 3 µ η 1 ≥ 1)then set h 1 ← – µelse set h 0 ← –1.

108

else
if (η 0 + 4 µ η 1 < –2)
then set h 1 ← – µ.
5. q 0 ← f 0 + h 0 .
6. q 1 ← f 1 + h 1 .
7. r 0 ← n – (s 0 + µ s 1 ) q 0 – 2s 1 q 1 .
8. r 1 ← s 1 q 0 – s 0 q 1 .
9. Set Q ← O.
10. P 0 ← P.
11. While ((r 0 ≠ 0) or (r 1 ≠ 0))
11.1 If (r 0 odd), then
11.1.1 set u ← 2 – (r 0 – 2 r 1 mod 4).
11.1.2 set r 0 ← r 0 – u.
11.1.3 if (u = 1), then set Q ← Q + P 0 .
11.1.4 if (u = –1), then set Q ← Q – P 0 .
11.2 Set P 0 ← τP 0 .
11.3 Set (r 0 , r 1 ) ← (r 1 + µr 0 /2, – r 0 /2).
Endwhile
12. Output Q.
D.5

Generation of Pseudo-Random Curves (Prime Case)

Let l be the bit length of p, and definev =  ( l – 1) /160
w = l – 160v – 1.
1. Choose an arbitrary 160-bit string s as the domain parameter seed.
2. Compute h = SHA-1(s).
3. Let h 0 be the bit string obtained by taking the w rightmost bits of h.
4. Let z be the integer whose binary expansion is given by the 160-bit string s.
5. For i from 1 to v do:

109

5.1 Define the 160-bit string s i to be binary expansion of the integer(z + i) mod (2 160).
5.2 Compute h i = SHA-1(s i ).
6. Let h be the bit string obtained by the concatenation of h 0 , h 1 , … , h v as follows:h = h 0 || h 1 || … || h v .
7. Let c be the integer whose binary expansion is given by the bit string h.
8. If ((c = 0 or 4c + 27 ≡ 0 (mod p))), then go to Step 1.
9. Choose integers a, b ∈GF(p) such thatc b2 ≡ a3 (mod p).
(The simplest choice is a = c and b = c. However, one may want to choose differently for
performance reasons.)
10. Check that the elliptic curve E over GF(p) given by y 2 = x3 + ax + b has suitable order. If
not, go to Step 1.
D.6

Verification of Curve Pseudo-Randomness (Prime Case)

Given the 160-bit domain parameter seed value s, verify that the coefficient b was obtained froms via the cryptographic hash function SHA-1 as follows.
Let l be the bit length of p, and definev =  ( l – 1) /160
w = l – 160v – 1.
1. Compute h = SHA-1(s).
2. Let h 0 be the bit string obtained by taking the w rightmost bits of h.
3. Let z be the integer whose binary expansion is given by the 160-bit string s.
4. For i = 1 to v do
4.1 Define the 160-bit string s i to be binary expansion of the integer(z + i) mod (2160 ).
4.2 Compute h i = SHA-1(s i ).
5. Let h be the bit string obtained by the concatenation of h 0 , h 1 , … , h v as follows:h = h 0 || h 1 || … || h v .
110

6. Let c be the integer whose binary expansion is given by the bit string h.
7. Verify that b2 c ≡ –27 (mod p).
D.7

Generation of Pseudo-Random Curves (Binary Case)Let:v =  (m – 1) /B
w = m – Bv.
1. Choose an arbitrary 160-bit string s as the domain parameter seed.
2. Compute h = SHA-1(s).
3. Let h 0 be the bit string obtained by taking the w rightmost bits of h.
4. Let z be the integer whose binary expansion is given by the 160-bit string s.
5. For i from 1 to v do:
5.1 Define the 160-bit string s i to be binary expansion of the integer(z + i) mod (2160 ).
5.2 Compute h i = SHA-1(s i ).
6. Let h be the bit string obtained by the concatenation of h 0 , h 1 , … , h v as follows:h = h 0 || h 1 || … || h v .
7. Let b be the element of GF(2m) which binary expansion is given by the bit string h.
8. Choose an element a of GF(2m).
9. Check that the elliptic curve E over GF(2m) given by y2 + xy = x3 + ax2 + b has suitableorder. If not, go to Step 1.
D.8

Verification of Curve Pseudo-Randomness (Binary Case)

Given the 160-bit domain parameter seed value s, verify that the coefficient b was obtained froms via the cryptographic hash function SHA-1 as follows.
Definev =  (m – 1) /160
w= m – 160v
1. Compute h = SHA-1(s).
2. Let h 0 be the bit string obtained by taking the w rightmost bits of h.
111

3. Let z be the integer whose binary expansion is given by the 160-bit string s.
4. For i = 1 to v do
4.1 Define the 160-bit string s i to be binary expansion of the integer (z + i) mod (2160 ).
4.2 Compute h i = SHA-1(s i ).
5. Let h be the bit string obtained by the concatenation of h 0 , h 1 , … , h v as follows:h = h 0 || h 1 || … || h v .
6. Let c be the element of GF(2m ) which is represented by the bit string h.
7. Verify that c = b.
D.9

Polynomial Basis to Normal Basis Conversion

Suppose that α is an element of the field GF(2m). Let p be the bit string representing α withrespect to a given polynomial basis. It is desired to compute n, the bit string representing α withrespect to a given normal basis. This is done via the matrix computationp Γ = n,where Γ is an m-by-m matrix with entries in GF(2). The matrix Γ, which depends only on thebases, can be computed easily given its second-to-last row. The second-to-last row for eachconversion is given the below.
Degree 163:
3 e173bfaf 3a86434d 883a2918 a489ddbd 69fe84e1
Degree 233:
0be 19b89595 28bbc490 038f4bc4 da8bdfc1 ca36bb05 853fd0ed
0ae200ce
Degree 283:
3347f17 521fdabc 62ec1551 acf156fb 0bceb855 f174d4c1 7807511c
9f745382 add53bc3
Degree 409:
0eb00f2 ea95fd6c 64024e7f 0b68b81f 5ff8a467 acc2b4c3 b9372843
6265c7ff a06d896c ae3a7e31 e295ec30 3eb9f769 de78bef5
Degree 571:
7940ffa ef996513 4d59dcbf e5bf239b e4fe4b41 05959c5d 4d942ffd
46ea35f3 e3cdb0e1 04a2aa01 cef30a3a 49478011 196bfb43 c55091b6
1174d7c0 8d0cdd61 3bf6748a bad972a4

112

Given the second-to-last row r of Γ, the rest of the matrix is computed as follows. Let β be theelement of GF(2m) whose representation with respect to the normal basis is r. Then the rows of
Γ, from top to bottom, are the bit strings representing the elements

β m–1, β m–2, …, β 2, β, 1with respect to the normal basis. (Note that the element 1 is represented by the all-1 bit string.)
Alternatively, the matrix is the inverse of the matrix described in Appendix D.10.
More details of these computations can be found in Annex A.7 of the IEEE Standard 1363-2000
standard.
D.10 Normal Basis to Polynomial Basis Conversion
Suppose that α is an element of the field GF(2m). Let n be the bit string representing α withrespect to a given normal basis. It is desired to compute p, the bit string representing α withrespect to a given polynomial basis. This is done via the matrix computationn Γ = p,where Γ is an m-by-m matrix with entries in GF(2). The matrix Γ, which depends only on thebases, can be computed easily given its top row. The top row for each conversion is given
below.
Degree 163:
7 15169c10 9c612e39 0d347c74 8342bcd3 b02a0bef
Degree 233:
149 9e398ac5 d79e3685 59b35ca4 9bb7305d a6c0390b cf9e2300
253203c9
Degree 283:
31e0ed7 91c3282d c5624a72 0818049d 053e8c7a b8663792 bc1d792e
ba9867fc 7b317a99
Degree 409:
0dfa06b e206aa97 b7a41fff b9b0c55f 8f048062 fbe8381b 4248adf92912ccc8 e3f91a24 e1cfb395 0532b988 971c2304 2e85708d
Degree 571:
452186b bf5840a0 bcf8c9f0 2a54efa0 4e813b43 c3d41496 06c4d27b
487bf107 393c8907 f79d9778 beb35ee8 7467d328 8274caeb da6ce05a
eb4ca5cf 3c3044bd 4372232f 2c1a27c4

113

Given the top row r of Γ, the rest of the matrix is computed as follows. Let β be the element of
GF(2 m) whose representation with respect to the polynomial basis is r. Then the rows of Γ, from
top to bottom, are the bit strings representing the elements

β, β 2 , β 2 , … , β 22

m–1

with respect to the polynomial basis.
Alternatively, the matrix is the inverse of the matrix described in Appendix D.9.
More details of these computations can be found in Annex A.7 of the IEEE Std 1363-2000
standard.

114

Appendix E: A Proof that v = r in the DSA(Informative)
The purpose of this appendix is to show that if M′ = M, r′ = r and s′ = s in the signature
verification, then v = r′. Let Hash be an approved hash function. The following result is needed.
Lemma: Let p and q be primes such that q divides (p – 1), let h be a positive integer less(p–1)/qthan p, and let g = (hmod p). Then (gq mod p) = 1, and if (m mod q) = (nmnmod q ), then (g mod p) = (g mod p).
Proof:
gqmod p = (h(p–1) / q mod p)q mod p= h(p – 1) mod p=1
by Fermat’s Little Theorem. Now let (m mod q) = (n mod q), i.e., m = (n + kq) for some integer
k. Then
gm mod p = gn + kq mod p= (gn gkq) mod p= ((gn mod p) (gq mod p)k) mod p= gn mod p,since (gq mod p) = 1.
Proof of the main result:
Theorem: If M′ = M, r′ = r, and s′ = s in the signature verification, then v = r′.
Proof:
w = (s′)–1 mod q = s–1 mod q
u1 = ((Hash(M ′))w) mod q = ((Hash(M))w) mod qu2 = ((r′)w) mod q = (rw) mod q.
x

Now y = (g mod p), so that by the lemma,v = ((gu1 yu2) mod p) mod q= ((gHash(M)w yrw) mod p) mod q= ((gHash(M)w gxrw) mod p) mod q= ((g(Hash(M) + xr)w) mod p) mod q.
115

Also:s = (k–1 (Hash(M) + xr)) mod q.
Hence:
w = (k (Hash(M) + xr)–1) mod q(Hash(M) + xr)w mod q = k mod q.
Thus, by the lemma:v = (gk mod p) mod q = r

116

Appendix F: Calculating the Required Number of Rounds of Testing
Using the Miller-Rabin Probabilistic Primality Test
(Informative)
F.1

The Required Number of Rounds of the Miller-Rabin Primality Tests

The ideas of paper [1] were applied to estimate p k,t , the probability that an odd k-bit integer thatpasses t rounds of Miller-Rabin (M-R) testing is actually composite. The probability p k ,t isunderstood as the ratio of the number of odd composite numbers of a binary length k that can be
expected to pass t rounds of M-R testing (with randomly generated bases) to the sum of thatvalue and the number of odd prime integers of binary length k. This is equivalent to assumingthat candidates selected for testing will be chosen uniformly at random from the entire set of odd
k-bit integers. Following Pomerance, et al., p k,t can be (over) estimated by the ratio of the
expected number of odd composite numbers of binary length k that will pass t rounds of M-R
testing (with randomly generated bases) to the total number of odd primes of binary length k.
From the perspective of a party charged with the responsibility of generating a k-bit prime, theobjective is to determine a value of t such that p k ,t is no greater than an acceptably small target
value pt arg et .
Using [1], it is possible to compute an upper bound for p k ,t as a function of k and t. From this,an upper bound can be computed for t as a function of k and p target , the maximum allowed
probability of accidentally generating a composite number. The following is an algorithm forcomputing t:
1. For t = 1, 2 … –log 2 (p target )/2



1.1 For M = 3, 4 … 2 k − 1 − 1
1.1.1

Compute p k ,t as in (2).

1.1.2

If p k,t ≤ p target
1.1.2.1

Accept t.

1.1.2.2

Stop.

(1)

In (1), k is the bit length of the candidate primes and (2) is as follows:8(π 2 − 6) k − 2 M m − ( m −1)t m
1
k − 2 − Mt .
pk ,t = 2.00743 ⋅ ln(2) ⋅ k ⋅ 2 2+
2 ∑2∑
 ( k −1)  3
m =3
j = 2  j + j 


2

−k

117

(2)

Using this expression for t, the following methodologies are used for testing the DSA and RSA
candidate primes.
F.2

Generating DSA Primes

For DSA, the maximum possible care must be taken when generating the primes p and q that areused for the domain parameters. The same primes p and q are used by many parties. This meansthat any weakness that these numbers may possess would affect multiple users. It also meansthat the primes are not generated very often; typically, an entire system uses the same set of
domain parameters for an extended period of time. Therefore, in this case, some additional careis called for.
With this in mind, it may be too optimistic to simply subject candidate primes to t rounds of M-R
testing, where the minimal acceptable value for t is determined according to (1) and (2) inAppendix F.1. This might be the case, for example, if there is a reason to doubt that theassumptions made in [1] have been satisfied during the process of selecting candidates forprimality testing. One may gain more confidence in the process by performing some additional(different) primality test(s) on the candidates that survive the M-R testing. As another option,
one could, of course, perform additional rounds of M-R testing. These considerations lead to thefollowing alternatives: either (A) use the number of rounds of M-R testing determined according
to (1) and (2) in Appendix F.1, and follow that with a single Lucas test (as recommended in ANS
X9.31), or (B) use a (much) more conservative approach when determining t (e.g., as described
below) and subject candidate primes to additional rounds of M-R testing.
One approach for strategy (B) would be to adopt the viewpoint of the majority of system users,
who have no part in generating the (supposed) prime, but who must rely upon its primality for
their security. Such parties may be concerned that the candidates for M-R testing have been
selected in a fashion that deviates significantly from the uniform distribution – which was
assumed when determining t according to (1) and (2) in Appendix F.1. In cases where theselection process could be unusually biased in some way, it is important to minimize theprobability that a composite number will survive testing. It can be shown that for any k-bit odd
composite number (regardless of how it was selected), the probability that it will pass t rounds of
M-R testing with randomly chosen bases is less than 4 − t (although this is not a particularly tightbound). Selecting t such that 4–t ≤ p target is equivalent to choosing t ≥ −log 2 (p target )/2. To ensurethat a composite number has a probability no greater than p target of surviving the M-R tests, thenumber of rounds can be set at t = –log 2 (p target )/2. Even if the method of selecting candidates
were so biased that it offered nothing but composite numbers for testing, it is reasonable toexpect that it would take at least 1/ p target attempts (which is greater than 4t ) before a composite
number would slip through the t-round M-R testing process.
WARNING: As the discussion above illustrates, care must be taken when using the phrase“error probability” in connection with the recommended number of rounds of M-R testing. The
probability that a composite number survives t rounds of Miller-Rabin testing is not the same as118

p k,t , which is the probability that a number surviving t rounds of Miller-Rabin testing iscomposite. Ordinarily, the latter probability is the one that should be of most interest to a party
responsible for generating primes, while the former may be more important to a party
responsible for validating the primality of a number generated by someone else. However, forsufficiently large k (e.g., k ≥ 51), it can be shown that p k,t ≤ 4–t under the same assumptions
concerning the selection of candidates as those made to obtain formula (2) in Appendix F.1 (see[1].) In such cases, t = –log 2 (p target )/2 rounds of Miller-Rabin testing can be used both ingenerating and validating primes, with p target serving as an upper bound on both the probability
that the generation process yields a composite number and the probability that a composite
number would survive an attempt to validate its primality.
Table C.1 in Appendix C.3 identifies the minimum values for t when generating the primes p andq for DSA using either strategy (A) or (B) above. To obtain the t values shown in the column
titled “M-R Tests Only”, the conservative strategy (B) was followed; those t values are sufficient
to validate the primality of p and q. The t values shown in the column titled “M-R Tests when
followed by One Lucas Test” result from following strategy (A) using computations (1) and (2)in Appendix F.1.
F.3

Generating Primes for RSA Signatures

When generating primes for the RSA signature algorithm, it is still very important to reduce theprobability of errors in the M-R testing procedure. However, since the (probable) primes areused to generate a user’s key pair, if a composite number survives the testing process, theconsequences of the error may be less dramatic than in the case of generating DSA domain
parameters; only one user’s transactions are affected, rather than a domain of users. Furthermore,
if the p or q value generated for some user is composite, the problem will not be undiscovered
for long, since it is almost certain that signatures generated by that user will not be verifiable.
Therefore, when generating the RSA primes p and q, it is sufficient to use the number of rounds
derived from (1) and (2) in Appendix F.1 as the minimum number of M-R tests to be performed.
However, if the definition of p k, t is not considered to be sufficiently conservative when testing p
and q, it is recommended that the t rounds of Miller-Rabin tests be followed by a single Lucas
test.
The lengths for p and q that are recommended for use in RSA signature algorithms are 512, 1024
and 1536 bits; recall that n = pq, so the corresponding lengths for n are 1024, 2048 and 3072 bits,
respectively. As currently specified in SP 800-57, Part 1, these lengths correspond to security
strengths of 80, 112 and 128 bits, respectively. Hence, it makes sense to match the number ofrounds of Miller-Rabin testing to the target error probability values of 2–80, 2–112, and 2–128. A
probability of 2–100 is included for all prime lengths, since this probability has often been used in
the past and may be acceptable for many applications.
When generating the RSA primes p and q with conditions, it is sufficient to use the value tderived from (1) and (2) as the minimum number of M-R tests to be performed when generating
119the auxiliary primes p 1 , p 2 , q 1 and q 2 . It is not necessary to use an additional Lucas test on thesenumbers. In the extremely unlikely event that one of the numbers p 1 , p 2 , q 1 or q 2 is composite,
there is still a high probability that the corresponding RSA prime (p or q) will satisfy therequisite conditions.
The sizes of p1 , p 2 , q1 , and q 2 were chosen to ensure that, for an adversary with significant but
not overwhelming resources, Lenstra’s elliptic curve factoring method [2] (against which there is
no protection beyond choosing large p and q) is a more effective factoring algorithm than either
the Pollard P–1 method [2], the Williams P+1 method [3] or various cycling methods [2]. For anadversary with overwhelming resources, the best all-purpose factoring algorithm is assumed tobe the General Number Field Sieve [2].
Tables C.2 and C.3 in Appendix C.3 specify the minimum number of rounds of M-R testing
when generating primes to be used in the construction of RSA signature key pairs.

120Appendix G: References[1] I. Damgard, P. Landrock, and C. Pomerance, C. “Average Case Error Estimates for theStrong Probable Prime Test,” Mathematics of Computation, v. 61, No, 203, pp. 177-194,
1993.
[2] A.J Menezes, P.C. Oorschot, and S.A. Vanstone. Handbook of Applied Cryptography. CRC
Press, 1996.
[3] H.C. Williams. “A p+1 Method of factoring”. Math. Comp. 39, 225-234, 1982.
[4] D.E. Knuth, The Art of Computer Programming, Vol. 2, 3rd Ed., Addison-Wesley, 1998,
Algorithm P, page 395.
[5] R. Baillie and S.S. Wagstaff Jr., Mathematics of Computation, V. 35 (1980), pages 1391 –
1417.

121